diff --git "a/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt" "b/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/79AzT4oBgHgl3EQf-f7W/content/tmp_files/load_file.txt" @@ -0,0 +1,931 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf,len=930 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='01936v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='PR] 5 Jan 2023 Clustering of large deviations in moving average processes: the long memory regime Arijit Chakrabarty Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Kolkata e-mail: arijit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='isi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='com and Gennady Samorodnitsky ∗ School of Operations Research and Information Engineering Cornell University e-mail: gs18@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='edu Abstract: We investigate how large deviations events cluster in the frame- work of an infinite moving average process with light-tailed noise and long memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The long memory makes clusters larger, and the asymptotic be- haviour of the size of the cluster turns out to be described by the first hitting time of a randomly shifted fractional Brownian motion with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' AMS 2000 subject classifications: Primary 60F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Keywords and phrases: large deviations, clustering, infinite moving av- erage, long memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Introduction We consider an infinite moving average process of the form Xn = ∞ � i=0 aiZn−i , n ≥ 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) where the noise variables (Zn : n ∈ Z) are assumed to bef i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' non-degenerate random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The noise distribution FZ is assumed have finite exponential moments: � R etz FZ(dz) < ∞ for all t ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) Furthermore, assuming that the noise is centred: � R z FZ(dz) = 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) ∗∗The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Research partially supported by NSF grant DMS-2015242 at Cornell University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Part of this work was performed when GS was visiting Department of Mathematics of National University of Singapore, whose hospitality is gratefully acknowl- edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 Chakrabarty and Samorodnitsky/Clustering of large deviations 2 the series defining the process in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) converges if and only if the coefficients a0, a1, a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' satisfy ∞ � j=0 a2 j < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) In this case (Xn) is a zero mean stationary ergodic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For ε > 0 we consider the sequence of large deviation events Ej(n, ε) = \uf8f1 \uf8f2 \uf8f3 1 n n+j−1 � i=j Xi ≥ ε \uf8fc \uf8fd \uf8fe , j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) By stationarity, each event Ej(n, ε) is equally rare, and we are interested in the cluster of these events that occur given that the event E0(n, ε) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In Chakrabarty and Samorodnitsky (2022) the short memory case was con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In this context, “short memory” corresponds to the case ∞ � n=0 |an| < ∞ and ∞ � n=0 an ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6) In this short memory case the conditional on E0(n, ε) law of the sequence � 1(Ej(n, ε), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=') converges weakly, as n → ∞, to the law of a sequence with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' finitely many non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the total number Dε of the non-zero en- tries turns out to scale as ε−2, and ε2Dε has an interesting weak limit as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We refer the reader to Chakrabarty and Samorodnitsky (2022) for details, and a minor technical condition required for the above statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In the present paper we are interested in the long memory case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For the mov- ing average processes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) “long memory” refers to the case when the coeffi- cients (aj) satisfy the square summability assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) but not the absolute summability assumption in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A typical assumption in this is (an) is regularly varying with exponent − α, 1/2 < α < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) see Samorodnitsky (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It turns out that, in this case (under certain technical assumptions, an example of which is below), the conditional on E0(n, ε) law of the sequence � 1(Ej(n, ε), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=') converges weakly, as n → ∞, to the degenerate probability measure δ(1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, once the event E0(n, ε) occurs, the events (Ej(n, ε)) become very likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to understand their structure we concentrate on the random variables In(ε) = inf {j ≥ 1 : Ej(n, ε) does not occur} , n ≥ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) and establish a weak limit theorem for this sequence under a proper scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Interestingly, the limit turns out to be the law of the first hitting time of a randomly shifted fractional Brownian motion with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The main result containing the above limit theorem and the technical as- sumptions it requires are in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The proof of the main theorem requires a long sequence of preliminary results, all of which are presented in that section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, some useful facts needed for the proofs are collected in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The assumptions and the main result Our result on clustering of large deviation events in the long memory case will require a number of assumptions that we state next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' First of all, we will replace the assumption of regular variation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) by the asymptotic power function assumption an ∼ n−α, 1/2 < α < 1, and is eventually monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) There is no doubt that the results of the paper hold under the more general regular variation assumption as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The extra generality will, however, require making an already highly technical argument even more so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The potentially resulting lack of clarity makes the added generality less valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The same is true about the eventual monotonicity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will need additional assumptions on the distribution of the noise variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will assume that some θ0 > 0, sup |θ|≤θ0 � ∞ −∞ t2 ���� � ∞ −∞ e(it+θ)z FZ(dz) ���� dt < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) Next, let σ2 Z = � R z2 FZ(dz) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) be the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote κ = the smallest integer > 4α − 1 2 − 2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) In other words, κ = � (1+2α)/(2−2α) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We assume that a generic noise variable Z satisfies EZi = EGi for 1 ≤ i ≤ κ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) where G ∼ N(0, σ2 Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is standard to verify that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) implies that the noise distri- bution has a twice continuously differentiable density fZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' One the other hand, a sufficient condition for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) is that the noise distribution has a four times continuously differentiable density fZ such that � ∞ −∞ eθ0|x| ���� di dxi fZ(x) ���� dx < ∞ for i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The moment equality assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) restricts how far the the noise distri- bution can be from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Note that in the range 1/2 < α < 5/8 we have κ = 2, in which case the assumption is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since κ ≥ 2 for all α ∈ (1/2, 1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) is implied by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 4 To state our main result, we need to introduce several key quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let β = 4 − 4α 3 − 2α ∈ (0, 1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6) and H = 3/2 − α ∈ (1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7) We denote by (BH(t) : t ≥ 0) the standard fractional Brownian motion with Hurst index H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' a zero mean Gaussian process with continuous paths and covariance function E (BH(s)BH(t)) = 1 2 � s2H + t2H − |s − t|2H� , s, t ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) If T0 is a standard exponential random variable independent of the fractional Brownian motion, then τε = inf � t ≥ 0 : BH(t) ≤ (2Cα)−1/2εt2H − (Cα/2)1/2σ2 Zε−1T0 � , ε > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='9) is an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' finite and strictly positive random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Here σ2 Z is the variance of the noise in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) and Cα = B(1 − α, 2α − 1) (1 − α)(3 − 2α) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10) with B(·, ·) the standard Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We are now in a position to state the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Assume the finite exponential moment condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2), that the coefficients satisfy the power-type condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1), the regularity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) and the moment equality condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for every ε > 0 the first non- occurrence times (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) satisfy P � n−βIn(ε) ∈ · ��E0(n, ε) � ⇒ P (τε ∈ ·) , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='11) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is worthwhile to observe that the limit law obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 depends on the noise distribution only through its variance σ2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This can be understood by noticing that in the long memory case considered in this paper we have Var(X1 + · · · + Xn) ≫ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, the events Ej(n, ε) should be viewed as moderate deviation events, not large deviation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It has been observed in many situations that moderate deviation events are influenced by the Gaussian weak limit of the quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' At the intuitive level, this explains why it is the variance of the process that appears in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For comparison, in the short memory case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6), we have Var(X1+· · ·+Xn) ∼ cn for some c > 0, the events Ej(n, ε) should be viewed as large deviation events, and their behaviour depends on much more than just the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' See Chakrabarty and Samorodnitsky (2022) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 5 We start on the road to proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 by establishing certain basic estimates that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote Aj = j � i=0 ai, j ∈ Z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='12) with the convention that a sum (or an integral) is zero if the lower limit exceeds the upper limit (so that Aj = 0 for j ≤ −1, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Sn = n−1 � i=0 Xi, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='13) and denote σ2 n = Var(Sn), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='14) In the sequel we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will denote by ϕZ(t) = log �� R etz FZ(dz) � , t ∈ R (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='15) the log-Laplace transform of a noise variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will frequently use the obvious facts ϕ is convex and ϕZ(x) ∼ x2σ2 Z/2, x → 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='16) and ϕ′ Z is continuous, nondecreasing and ϕ′ Z(x) = xσ2 Z + O(x2), x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17) We will write Gθ for the probability measure obtained by exponentially tilting the law FZ by θ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, Gθ(dz) = � EeθZ�−1eθzFZ(dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18) It is clear that, as θ → 0, � R z Gθ(dz) ∼ θσ2 Z, ���� � R z Gθ(dz) − θσ2 Z ���� = O(θ2) and = O(|θ|3) if κ ≥ 3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) � R |z|k Gθ(dz) → � R |z|k F(dz), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='. Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Asymptotically we have Aj ∼ (1 − α)−1j1−α, j → ∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) and σ2 n ∼ Cασ2 Zn3−2α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) Chakrabarty and Samorodnitsky/Clustering of large deviations 6 Furthermore, for any t > 0, as n → ∞, [nβt] � i=0 (Ai − Ai−n)2 ∼ K1t3−2αn4−4α , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22) and n � i=n−[nβt]+1 (Ai − Ai−n)2 ∼ n+[nβt] � i=n+1 (Ai − Ai−n)2 ∼ (1 − α)−2n2−2α+βt , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='23) with K1 = (1 − α)−2(3 − 2α)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24) Finally, for any t > 0, as n → ∞, σ2 Z σ2n ∞ � i=0 (Ai − Ai−n) � Ai+[nβt] − Ai+[nβt]−n � = 1−n1−2αt3−2α(1+o(1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) is, of course, an immediate consequence of the assump- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21), first note that Rn = Cov(X0, Xn) ∼ σ2 Z ∞ � j=1 j−α(j + n)−α ∼ n1−2ασ2 Z � ∞ 0 x−α(1 + x)−α dx = Cασ2 Z(1 − α)(3 − 2α)n1−2α as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, σ2 n = n−1 � i=−(n−1) (n − |i|)R|i| ∼ 2Cασ2 Z(1 − α)(3 − 2α) n−1 � i=0 (n − i)i1−2α ∼ 2Cασ2 Z(1 − α)(3 − 2α)n3−2α � 1 0 (1 − x)x1−2α dx = Cασ2 Zn3−2α , which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Next, for a fixed t > 0 and large n, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20) and the fact that β < 1, [nβt] � i=0 (Ai − Ai−n)2 = [nβt] � i=0 A2 i ∼ (1 − α)−2 [nβt] � i=1 i2−2α ∼ K1 � nβt �3−2α , proving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Similarly, n � i=n−[nβt]+1 (Ai − Ai−n)2 ∼ n � i=n−[nβt]+1 A2 n ∼ (1 − α)−2nβ+2−2αt , Chakrabarty and Samorodnitsky/Clustering of large deviations 7 showing the first equivalence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='23) and the second equivalence can be shown in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25), we start by writing Sn = ∞ � j=0 (Aj − Aj−n)Zn−1−j , n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26) so that σ2 n = σ2 Z ∞ � j=0 (Aj − Aj−n)2 , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='27) Therefore, for large n, σ2 n σ2 Z − ∞ � i=0 (Ai − Ai−n)(Ai+[nβt] − Ai+[nβt]−n) = 1 2 \uf8ee \uf8f0 [nβt]−1 � i=0 (Ai − Ai−n)2 + ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 \uf8f9 \uf8fb = 1 2 �n−1 � i=0 � Ai − Ai−[nβt] �2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) + ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='20), n−1 � i=0 � Ai − Ai−[nβt] �2 ∼ (1 − α)−2 n−1 � i=1 � i1−α − (i − [nβt])1−α + �2 ∼ n4−4αt3−2α(1 − α)−2 � ∞ 0 � y1−α − (y − 1)1−α + �2 dy as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) with H = 3/2 − α, � ∞ 0 � y1−α − (y − 1)1−α + �2 dy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='29) = [(3 − 2α) (1 − α)]−1 sin(πα) π Γ(2α − 1)Γ(2 − α)2 = 1 − α 3 − 2αB (2α − 1, 1 − α) = (1 − α)2Cα, so n−1 � i=0 � Ai − Ai−[nβt] �2 ∼ Cαt3−2αn4−4α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='30) Chakrabarty and Samorodnitsky/Clustering of large deviations 8 Since ∞ � i=n � Ai − Ai−[nβt] �2 = O � n2β ∞ � i=n i−2α � = O � n2β+1−2α� = o � n4−4α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='31) we conclude also that ∞ � i=0 � Ai − Ai−[nβt] �2 ∼ Cαt3−2αn4−4α, n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='32) It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='31) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='32) that ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 = ∞ � j=0 � −Aj + Aj−[nβt] + � Aj+n − Aj+n−[nβt] ��2 ∼ Cαt3−2αn4−4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In combination with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='30) we obtain σ2 n σ2 Z − ∞ � i=0 (Ai − Ai−n)(Ai+[nβt]Ai+[nβt]−n) ∼ Cαt3−2αn4−4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Dividing both sides by σ−2 Z σ2 n and appealing to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now consider certain large deviations of the partial sum Sn under a change of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With an eye towards a subsequent application, we allow the partial sum, given in the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26), to be “corrupted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and t ≥ 0 we define ξ1 n(t) = [nβt] � i=1 (Ai − Ai−n) Zn−i−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33) ξ2 n(t) = n−1 � i=n−[nβt] (Ai − Ai−n) Zn−i−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) ξ3 n(t) = n+[nβt] � i=n+1 (Ai − Ai−n) Zn−i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix t1, t2, t3 > 0 and denote ¯Sn = Sn − 3 � i=1 ξi n(ti), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='36) Let (γn), (θn) and (ηn) be real sequences satisfying γn = o � n3/2−α� , θn = o � n−(1−α)� , 1 ≪ ηn ≪ n1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 9 If ˜Sn is a random variable with the law P � ˜Sn ∈ dx � = � E(eθn ¯Sn) �−1 eθnxP � ¯Sn ∈ dx � , n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='37) then for all x ∈ R and h > 0, P � ηnσ−1 n � ˜Sn − E( ˜Sn) + γn � ∈ [x, x + h] � ∼ η−1 n (2π)−1/2h, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) Furthermore, sup n≥1, x∈R ηnP � ηnσ−1 n ˜Sn ∈ [x, x + 1] � < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables such that the law of ˜Zni is G(Ai−Ai−n)θn in the notation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for large n, ˜Sn d= A0 ˜Zn0 + (An − A0) ˜Znn + n−[nβt2]−1 � i=[nβt1]+1 Ai ˜Zni + ∞ � i=n+[nβt3]+1 (Ai − Ai−n) ˜Zni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='40) The proof applies to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='40) the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) in the appendix, with n = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For any z ∈ R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='41) ≤ Cu � Var( ˜Sn) �−3/2 ∞ � i=0 |Ai − Ai−n|3E � | ˜Zni − E ˜Zni|3� , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is immediate from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1) that sup i≥0 |Ai − Ai−n| = O(n1−α) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) so that lim n→∞ θn sup i≥0 |Ai − Ai−n| = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) that E ˜Zni → 0, Var( ˜Zni) → σ2 Z, E � | ˜Zni − E ˜Zni|3� → � ∞ −∞ |z3| FZ(dz) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) uniformly in i as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since it is an elementary conclusion from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that for any κ > 1/α, ∞ � i=0 |Ai − Ai−n|κ = O � nκ+1−κα� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44) Chakrabarty and Samorodnitsky/Clustering of large deviations 10 it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='41) that sup z∈R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� = O � n4−3α � Var( ˜Sn) �−3/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) again we see that Var( ˜Sn) ∼ σ2 n − 3 � i=1 Var � ξi n(ti) � ∼ Cασ2 Zn3−2α, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='45) with the second equivalence following from various claims in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, sup z∈R ����P � ˜Sn − E( ˜Sn) ≤ z � Var( ˜Sn) � − Φ(z) ���� = O(n−1/2) = o � η−1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='46) Therefore, for x ∈ R and h > 0, as n → ∞, P � ηnσ−1 n � ˜Sn − E( ˜Sn) + γn � ∈ [x, x + h] � = o � η−1 n � + � R 1 � Var( ˜Sn)−1/2(xη−1 n σn − γn) ≤ z ≤ Var( ˜Sn)−1/2((x + h)η−1 n σn − γn) � φ(z) dz, where φ is the standard normal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The assumptions on γn and ηn along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='45) imply that the integration interval shrinks towards the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, the integral above is asymptotically equivalent to η−1 n φ(0)h, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Boundedness of φ in the above integral establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now look more closely at the processes defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma describes the limiting distribution of their increments under the same change of measure as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that θn ∈ R satisfies θn = o � n−(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix 0 ≤ s < t and consider random variables with the laws P(Uni ∈ dx) = cnieθnxP � ξi n(t) − ξi n(s) ∈ dx � , i = 1, 2, 3, n ≥ 1 , with appropriate cni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then, as n → ∞, n−(2−2α) (Un1 − E(Un1)) ⇒ N � 0, K1σ2 Z � t3−2α − s3−2α�� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='47) where K1 is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24), and for i = 2, 3, n−(1−α+β/2) (Uni − E(Uni)) ⇒ N � 0, (1 − α)−2σ2 Z(t − s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='48) Chakrabarty and Samorodnitsky/Clustering of large deviations 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For large n, Un1 d= [nβt] � i=[nβs]+1 Ai ˜Zni with ( ˜Zni) as in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That is, Un1 − E(Un1) is the sum of independent zero mean random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22), Var(Un1) ∼ σ2 Z [nβt] � i=[nβs]+1 A2 i ∼ K1σ2 Zn4−4α � t3−2α − s3−2α� , and a similar calculation using the third moment bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='43) verifies the Lindeberg conditions of the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='47) follows, and the calculations for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='48) are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Consider the overshoot defined by T ∗ n = Sn − nε, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='49) Conditionally on the event E0 = E0(n, ε) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) the overshoot is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma is a joint weak limit theorem for the joint law of the overshoot and the processes defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='33), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The joint law is computed conditionally on E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ζn = nε/σ2 n, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='50) Conditionally on E0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' as n → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � ζnT ∗ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' \uf8eb \uf8edn2α−2 \uf8eb \uf8edξ1 n(t) − [nβt] � i=1 Ai � ∞ −∞ z GζnAi(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � nα−β/2−1 \uf8eb \uf8edξ2 n(t) − n−1 � i=n−[nβt] Ai � ∞ −∞ z GζnAi(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � nα−β/2−1 \uf8eb \uf8edξ3 n(t) − n+[nβt] � i=n+1 (Ai − Ai−n) � ∞ −∞ z Gζn(Ai−Ai−n)(dz) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 �� ⇒ � T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � K1/2 1 σZB1(t3−2α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � (1 − α)−1σZB2(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' � (1 − α)−1σZB3(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ≥ 0 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' in finite dimensional distributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' where T0 is a standard exponential random variable independent of independent standard Brownian motions B1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' B2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' and B3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' K1 is the constant in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='24) and Gθ is the exponentially tilted law in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote ψn(s) = σ2 n n2 log E � exp � s n σ2n Sn �� = σ2 n n2 ∞ � j=0 ϕZ � σ−2 n n(Aj − Aj−n)s � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='51) where the second equality follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) we see that lim n→∞ ψn(s) = s2/2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='52) uniformly for s in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='51) can be differ- entiated term by term, and it follows by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) that lim n→∞ ψ′ n(s) = s, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='53) also uniformly on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since ψ′ n is increasing and continuous, for large n there exists a unique τn > 0 such that ψ′ n(τn) = ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='54) It is immediate that τn → ε as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denoting θn = σ−2 n nτn, n ≥ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='55) we have � E � eθnSn��−1 E � SneθnSn� = nε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='56) Fix k ≥ 1 and for each i = 1, 2, 3 fix points 0 = ti0 < ti1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' < tik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote ¯Sn = Sn − 3 � i=1 ξi n(tik), n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Unij, n ≥ 1, i = 1, 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, ˜Sn, n ≥ 1 be independent random variables, with P (Unij ∈ dx) = � E � eθn(ξi n(tij)−ξi n(ti j−1))��−1 eθnxP � ξi n(tij) − ξi n(ti j−1) ∈ dx � , and P � ˜Sn ∈ dx � = � E � eθn ¯Sn��−1 eθnxP � ¯Sn ∈ dx � for n ≥ 1, i = 1, 2, 3 and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let µnij = E (Unij) , µn = E( ˜Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='57) Chakrabarty and Samorodnitsky/Clustering of large deviations 13 It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='56) that µn + 3 � i=1 k � j=1 µnij = nε, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='58) Let t > 0 and (αij) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have P �� T ∗ n > tσ2 n/nε � ∩ � k� j=1 � n2α−2 � ξ1 n(t1j) − ξ1 n(t1 j−1) − µn1j � > α1j �� ∩ � � 2≤i≤3, 1≤j≤k � nα−β/2−1 � ξi n(tij) − ξi n(ti j−1) − µnij � > αij ��� = � R3k+1 1 � x > nε + tσ2 n/nε − 3 � i=1 k � j=1 sij � 1 � s1j > n2−2αα1j + µn1j , 1 ≤ j ≤ k � 1 � sij > n1−α+β/2αij + µnij , i = 2, 3 , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' k � P( ¯Sn ∈ dx) 3 � i=1 k � j=1 P � ξi n(tij) − ξi n(ti j−1) ∈ dsij � = � R3k+1 1 � x > nε + tσ2 n/nε − 3 � i=1 k � j=1 sij � 1 � s1j > n2−2αα1j + µn1j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 ≤ j ≤ k � 1 � sij > n1−α+β/2αij + µnij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' i = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1 ≤ j ≤ k � exp � −θnx − θn 3 � i=1 k � j=1 sij � P � ˜Sn ∈ dx � E � eθnSn� 3 � i=1 k � j=1 P(Unij ∈ dsij) = cn � R3k 1 � min i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='j (uij − αij) > 0 � k � j=1 P � n2α−2� Un1j − µn1j � ∈ du1j � 3 � i=2 k � j=1 P � nα−β/2−1� Unij − µnij � ∈ duij � � R e−z1 � z > tθnσ2 n/nε � P � θn � ˜Sn − µn + γn(u11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) � ∈ dz � , Chakrabarty and Samorodnitsky/Clustering of large deviations 14 with cn = e−θnnεE � eθnSn� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='59) and γn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) = n2−2α k � j=1 u1j + n1−α+β/2 3 � i=2 k � j=1 uij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let θn be as above and ηn = σnθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1, we introduce the notation fn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) =ηn � ∞ 0 e−z1 � z > tθnσ2 n/nε � P � θn � ˜Sn − µn + γn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) � ∈ dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix (uij) and let u(n) ij → uij as n → ∞ for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let us denote γn = γn � u(n) 11 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u(n) 3k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With θn and ηn already defined, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2 with this γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is elementary to check that the hypothesis of the lemma are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since tθnσ2 n/nε → t, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='38) that for all fixed T > t, � R e−z1 � tθnσ2 n/nε < z ≤ T � P � θn � ˜Sn − µn + γn � ∈ dz � ∼ η−1 n (2π)−1/2 � T t e−z dz, and if follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) that lim T →∞ lim sup n→∞ ηn � R e−z1 � z > T � P � θn � ˜Sn − µn + γn � ∈ dz � = 0 , showing that lim n→∞ fn � u(n) 11 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u(n) 3k � = (2π)−1/2e−t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Another application of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='39) implies that sup {uij}⊂R fn(u11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , u3k) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3 and bounded convergence theorem that E � f � n2α−2(Un11 − µn11), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , nα−β/2−1(Un3k − µn3k) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) 1 � n2α−2(Un1j − µn1j) > α1j, nα−β/2−1(Unij − µnij) > αij, i = 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, �� →(2π)−1/2P (T0 > t , Gij > αij for all i, j) , with T0 standard exponential and (Gij : i = 1, 2, 3, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k) independent zero mean Gaussian random variables, independent of T0, with Var(G1j) = K1σ2 Z � t3−2α 1j − t3−2α 1 j−1 � , 1 ≤ j ≤ k , Chakrabarty and Samorodnitsky/Clustering of large deviations 15 and for i = 2, 3, Var(Gij) = (1 − α)−2σ2 Z(tij − ti ,j−1), 1 ≤ j ≤ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A simple way to verify the convergence above is to appeal to the Skorohod representation and replace the weak convergence in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3 by the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Notice that using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) with t = 0 and αij = −∞ for all i, j tells us that P(E0) ∼ (2π)−1/2cn/ηn = (2π)−1/2e−θnnεE � eθnSn� /(σnθn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61) Dividing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='60) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61) gives us the statement of the lemma apart from a possibly different centring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to complete the proof, it suffices to show that as n → ∞, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' , k, µn1j = [nβt1j] � i=[nβt1j−1]+1 Ai � ∞ −∞ z GζnAi(dz) + o � n2−2α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='62) µn2j = n−[nβtnj−1] � i=n−[nβtnj] Ai � ∞ −∞ z GζnAi(dz) + o � n1+β/2−α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) µn3j = n+[nβtnj] � i=n+[nβtnj−1] (Ai − Ai−n) � ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � n1+β/2−α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='64) For simplicity of notation we prove these statements for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For θn as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='55), let ( ˜Zni, n ≥ 1, i ≥ 0) be a collection of independent random variables such that the law of ˜Zni is G(Ai−Ai−n)θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since both θnAi and ζnAi converge to zero uniformly in i ≤ nβt11, we can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) to write µn11 = [nβt11] � i=1 AiE � ˜Zni � = [nβt11] � i=1 Ai � ∞ −∞ z GθnAi(dz) = [nβt11] � i=1 Ai � ∞ −∞ z GζnAi(dz) + o \uf8eb \uf8edζn [nβt11] � i=1 A2 i \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='22) that ζn [nβt11] � i=1 A2 i = o � n2−2α� , and we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='62) (for j = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) with j = 1 we notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='17), E � ˜Zni � = θn(Ai − Ai−n)σ2 Z + O � θ2 n(Ai − Ai−n)2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='65) Chakrabarty and Samorodnitsky/Clustering of large deviations 16 uniformly in i ≥ 0, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, µn21 = σ2 Zσ−2 n nτn n−1 � i=n−[nβt21] A2 i + O \uf8eb \uf8edθ2 n n−1 � i=n−[nβt21] A3 i \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that θ2 n n−1 � i=n−[nβt21] A3 i = O � nα+β−1� = o � n1−α+β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, µn21 = σ2 Zσ−2 n nτn n−1 � i=n−[nβt21] A2 i + o � n1−α+β/2� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='66) and, similarly, n−1 � i=n−[nβt21] Ai � ∞ −∞ z GζnAi(dz) = σ2 Zζn n−1 � i=n−[nβt21] A2 i + o � n1−α+β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Another appeal to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 shows that for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) we only need to argue that τn = ε + o � n1−α−β/2� , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='67) However, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19), ψ′ n(s) = s + O \uf8eb \uf8ednσ−4 n ∞ � j=0 (Aj − Aj−n)3 \uf8f6 \uf8f8 , uniformly for s in compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44), we obtain ε = ψ′ n(τn) = τn + O \uf8eb \uf8ednσ−4 n ∞ � j=0 (Aj − Aj−n)3 \uf8f6 \uf8f8 = τn + O(nα−1) = τn + o � n1−α−β/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='67) and, hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='63) with j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='64) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' None of the statements proved so far required the additional assumptions stated at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' These assumptions start to play a role now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 17 The next several lemmas require additional notation designed to focus on the contribution of individual noise variables on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and i, j ≥ 0, i ̸= j, we set S′ n(i) = Sn − (Ai − Ai−n)Zn−i−1 , S′ n(i, j) = Sn − (Ai − Ai−n)Zn−i−1 − (Aj − Aj−n)Zn−j−1 , and, with ζn given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='50), we let ˆSn, ˆSni, ˆSn(i, j) be random variables with distributions P( ˆSn ∈ ds) ∝ eζnsP(Sn ∈ ds) , P( ˆSn(i) ∈ ds) ∝ eζnsP(S′ n(i) ∈ ds) , P( ˆSn(i, j) ∈ ds) ∝ eζnsP(S′ n(i, j) ∈ ds) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Denote the characteristic functions of σ−1 n ( ˆSn − nε), σ−1 n ( ˆSn(i) − nε) and σ−1 n ( ˆSn(i, j) − nε) by φn, φni and φnij, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For µ ∈ R and σ ≥ 0 we denote by φG(µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ·) the characteristic function of N(µ, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let κ be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) and assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then the following statements hold uniformly in t ∈ R: |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| = O � n1/2−κ(1−α)(1 + |t|)κ+1� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='68) sup i≥0 ��φni(t) − φG � σ−1 n nε(λni − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ��� = O � n1/2−κ(1−α)(1 + |t|)κ+1� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) sup i,j≥0 i̸=j ��φnij(t) − φG � σ−1 n nε(λnij − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λnij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t ��� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='70) = O � n1/2−κ(1−α)(1 + |t|)κ+1� , where for n ≥ 1 and i, j ≥ 0, i ̸= j, we set λni = 1 − σ2 Z σ2n (Ai − Ai−n)2, λnij = 1 − σ2 Z σ2n � (Ai − Ai−n)2 + (Aj − Aj−n)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is an elementary conclusion from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) that, for each 1 ≤ i ≤ κ, �� R eδz Fz(dz) �−1 � R zieδz Fz(dz) = σi ZE � (G + δσZ)i� + O � |δ|κ−i+1� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='71) as δ → 0, where G is a standard Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let ( ˆZni : n ≥ 1, i ≥ 0) be a family of independent random variables with each ˆZni ∼ G(Ai−Ai−n)ζn, so that for n ≥ 1 and i, j ≥ 0, i ̸= j we have ˆSn d= ∞ � k=0 (Ak − Ak−n) ˆZnk , Chakrabarty and Samorodnitsky/Clustering of large deviations 18 ˆSn(i) d= � k∈{0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='}\\{i} (Ak − Ak−n) ˆZnk , ˆSn(i, j) d= � k∈{0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='}\\{i,j} (Ak − Ak−n) ˆZnk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let now (Gni : n ≥ 1, i ≥ 0) be a collection of independent random variables, also independent of ( ˆZni : n ≥ 1, i ≥ 0), where Gni ∼ N � (Ai − Ai−n)ζnσ2 Z , σ2 Z � , for all n ≥ 1, i ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='71) can be reformulated as E � ˆZi nj � − E � Gi nj � = O � |Aj − Aj−n|κ−i+1n−2(1−α)(κ−i+1)� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='72) uniformly in j ≥ 0 and 1 ≤ i ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For a fixed t ∈ R we use telescoping to write ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n ∞ � j=0 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe − E exp � i � tσ−1 n ˆSn �� ������ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='73) ≤ ∞ � j=0 ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe −E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix j ≥ 0 and denote U = tσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 , V = tσ−1 n (Aj − Aj−n)Gnj , so that by expanding in the Taylor series around U, E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = Eei(U+V ) = κ � m=0 im m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='E (V m) EeiU + R1 , with |R1| ≤ E(|V |κ+1)/(κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='. Similarly, E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = κ � m=0 im m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='E (W m) EeiU + R2 , Chakrabarty and Samorodnitsky/Clustering of large deviations 19 with |R2| ≤ E(|W|κ+1)/(κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=', where W = (Aj − Aj−n) ˆZnj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We conclude that ������ E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j−1 � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe −E exp \uf8f1 \uf8f2 \uf8f3i \uf8eb \uf8edtσ−1 n \uf8eb \uf8ed j � k=0 (Aj − Aj−n) ˆZnj + ∞ � k=j+1 (Aj − Aj−n)Gnj \uf8f6 \uf8f8 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe ������ ≤ κ � i=1 |t|i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' ���(Aj − Aj−n)iσ−i n E � ˆZi nj − Gi nj ���� + |t|κ+1 (κ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' |Aj − Aj−n|κ+1 σ−(κ+1) n E � |Gnj|κ+1 + | ˆZnj|κ+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='74) Note that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='44) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1, σ−(κ+1) n ∞ � j=0 |Aj − Aj−n|κ+1 E � |Gnj|κ+1 + | ˜Znj|κ+1� = O � n−(κ−1)/2� = o � n1/2−κ(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For 1 ≤ i ≤ κ we use, in addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='72) to write σ−i n ∞ � j=0 ���(Aj − Aj−n)iE � ˜Zi nj − Gi nj ���� = O � n−κ(1−α)+α−i(α−1/2)� = O � n1/2−κ(1−α)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Putting these bounds into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='74) we obtain E � eιtσ−1 n ˜Sn� = φG � σ−1 n nε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t � + O � n1/2−κ(1−α) � 1 + |t|κ+1�� uniformly for t ∈ R, which is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The argument for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='70) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2), for large n, the random variables σ−1 n ( ˆSn − nε), σ−1 n ( ˆSn(i) − nε) and σ−1 n ( ˆSn(i, j) − nε) have densities which we denote by fn, fni and fnij, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then for large n, the densities fni and fnij are twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, as n → ∞, fni(0) = (2π)−1/2 + o � n1−2α� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='75) f ′ ni(0) = o � n1/2−α� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) Chakrabarty and Samorodnitsky/Clustering of large deviations 20 uniformly in i, and for some n0 ∈ N, sup {|f ′′ ni(x)| : n ≥ n0, i ≥ 0, x ∈ R} < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='77) All three statements also hold if fni is replaced by fnij, i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, as n → ∞, sup x∈R ���fn(x) − (2π)−1/2e−x2/2��� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We start with the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) which would follow from the inversion formula for densities once it is shown that � ∞ −∞ |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| dt = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4), � log n − log n |φn(t) − φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t)| dt = O � n1/2−κ(1−α)(log n)κ+2� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, � [− log n,log n]c φG(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t) dt = O � e−(log n)2/2� = o � n1−2α� , Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) will follow once we show that � [− log n,log n]c |φn(t)| dt = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='79) With ( ˆZni : n ≥ 1, i ≥ 0) as above, we set Uni = σ−1 n (Ai − Ai−n) � ˆZni − E( ˆZni) � , n ≥ 1, i ≥ 0 , so that |φn(t)| = ∞ � i=0 ��E � eιtUni��� , n ≥ 1, t ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='80) Set H(x, t) = �� ∞ −∞ exzfZ(z) dz �−1 � ∞ −∞ e(x+ιt)zfZ(z) dz, (x, t) ∈ R2 , which is a characteristic function for any fixed x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A consequence of that is ∂|H(x, t)|/∂t|t=0 ≤ 0 for any x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, ∂2 ∂t2 |H(0, t)| ��� t=0 = −σ2 Z < 0 Chakrabarty and Samorodnitsky/Clustering of large deviations 21 and by continuity of the second partial derivative we conclude that there is δ0 > 0 such that ∂2 ∂t2 |H(x, t)| ��� < 0 whenever 0 ≤ |t|, |x| ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' That means we also have ∂ ∂t|H(x, t)| ��� ≤ 0 whenever 0 ≤ |t|, |x| ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) We may and will choose δ0 ∈ (0, θ0], with θ0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) we can appeal to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='3) to conclude that lim t→∞ sup |x|≤δ0 |H(x, t)| = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, there is M > 0 large enough so that sup t>M,|x|≤δ0 |H(x, t)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by continuity of H and compactness we have sup δ0≤t≤M,|x|≤δ0 |H(x, t)| < 1, it follows that η = sup t≥δ0,|x|≤δ0 |H(x, t)| < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The continuity argument also shows that there is δ1 ∈ (0, δ0] such that min |x|≤δ0 |H(x, δ1)| ≥ η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, for |x| ≤ δ0 and 0 ≤ t ≤ δ1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) implies that |H(x, t)| ≥ |H(x, δ1)| ≥ η ≥ sup s≥δ0 |H(x, s)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='81) we also have |H(x, t)| = sup s∈[t,δ0] |H(x, s)| , we conclude that |H(x, t)| = sup s≥t |H(x, s)|, |x| ≤ δ0, 0 ≤ t ≤ δ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='82) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='80) |φn(t)| ≤ ��E(eιtUnn) �� n−1 � i=[n/2] ��E(eιtUni) �� = ��E(eιtUnn) �� n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='83) Chakrabarty and Samorodnitsky/Clustering of large deviations 22 It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1 that there exists s0 > 0 such that for all n large enough, Ai ≥ s0σnn−1/2, [n/2] ≤ i ≤ n − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, for n large enough and t ≥ log n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='82) implies that n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� ≤ n−1 � i=[n/2] ���H � ζnAi, s0n−1/2 log n ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since any partial derivative of H is bounded on a compact set, we can use the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4) to conclude that there exists s1 > 0 such that sup |x|≤δ0 |H(x, t)| ≤ (1 − s1t2)1/2, 0 ≤ t ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Thus, there is s2 > 0 such that for all large n and all t ≥ log n we have n−1 � i=[n/2] ��H � ζnAi, σ−1 n Ait ��� ≤ � 1 − s2 0s1n−1(log n)2�n/4 = O � e−s2(log n)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Using this bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='83), and appealing to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) we obtain � ∞ log n |φn(t)| dt = O � e−s2(log n)2� � ∞ log n ��E � eitUnn��� dt = O � n1/2e−s2(log n)2� = o � n1−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since we can switch from t to −t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='79) follows, which establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A similar calculation with the aid of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) shows that fni(0) = (2πλni)−1/2 exp � −σ−2 n n2ε2(λni − 1)2/2λni � + o � n1−2α� , uniformly in i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since λni − 1 = O(1/n) uniformly in i ≥ 0, it follows that λ−1/2 ni exp � −σ−2 n n2ε2(λni − 1)2/2λni � = 1 + O � n−1 + σ−2 n � = 1 + o � n1−2α� , uniformly for i ≥ 0, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='77) we write f ′′ nk(x) = −(2π)−1/2 � ∞ −∞ e−itxt2φnk(t) dt and repeat the arguments used above in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78), applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='69) and the full force of the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) we use the identity f ′ nk(0) = −i(2π)−1/2 � ∞ −∞ tφnk(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 23 Since ���� � ∞ −∞ t φG � σ−1 n nε(λnk − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' λnk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' t � dt ���� = O � σ−1 n � = o � n1/2−α� , uniformly in k ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='76) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The arguments with fnij replacing fni are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma tackles certain expectations conditionally on E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' its state- ment should be compared to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then E (Zn−i−11(E0)) = Kn �� ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � ζ−1 n σ−2 n |Ai − Ai−n| �� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) and E (Zn−i−1Zn−j−11(E0)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) = Kn �� ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Ai−Ai−n)(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � � , n → ∞, uniformly for i, j ≥ 0 with i ̸= j, where Kn = (2π)−1/2ζ−1 n σ−1 n e−nεζnE � eζnSn� , n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='86) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We only prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) is similar and easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Write E (Zn−i−1Zn−j−11(E0)) = � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) P (S′ n(i, j) ≥ nε − (Ai − Ai−n)z1 − (Aj − Aj−n)z2) = σ−1 n E � eζnS′ n(i,j)� � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) � ∞ nε−(Ai−Ai−n)z1−(Aj−Aj−n)z2 fnij � s − nε)/σn � e−ζns ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We adopt the convention � b a ≡ − � a b , and denote cnij = ζ−1 n σ−1 n e−nεζnE � eζnS′ n(i,j)� = Kn(2π)1/2 �� ∞ −∞ eζn(Ai−Ai−n)zFZ(dz) � ∞ −∞ eζn(Aj−Aj−n)zFZ(dz) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 24 Changing the variable and using the fact that EZ = 0, we obtain E (Zn−i−1Zn−j−11(E0)) = cnij � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) � ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx = cnij � ∞ −∞ z1 FZ(dz1) � ∞ −∞ z2 FZ(dz2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) �� ζn(Ai−Ai−n)z1+ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx − � ζn(Ai−Ai−n)z1 0 exfnij � −x/(σnζn) � dx − � ζn(Aj−Aj−n)z2 0 exfnij � −x/(σnζn) � dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For fixed z1, z2 ∈ R, the expression inside the square brackets can be rewritten as � eζn(Ai−Ai−n)z1 − 1 � � ζn(Aj−Aj−n)z2 0 ex fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx + � ζn(Aj−Aj−n)z2 0 ex � fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � − fnij � −x/(σnζn) � � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By Taylor’s theorem, fnij � − x + ζn(Ai − Ai−n)z1 σnζn � = fnij(0) − x + ζn(Ai − Ai−n)z1 σnζn f ′ nij(0) +O �(x + ζn(Ai − Ai−n)z1)2 σ2nζ2n ∥f ′′ nij∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 25 Using this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='42), straightforward algebra gives us � ζn(Aj−Aj−n)z2 0 exfnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx = fnij(0) � eζn(Aj−Aj−n)z2 − 1 � + O � eζn|Aj−Aj−n||z2|� |f ′ nij(0)|σ−1 n ζnn1−α|Aj − Aj−n||z2| � |z1| + |z2| � + ∥f ′′ nij∥∞σ−2 n ζnn2−2α|Aj − Aj−n||z2| � |z1| + |z2| �2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The obvious inequality |ex − 1| ≤ |x|e|x| for x ∈ R along with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6 now show that � eζn(Ai−Ai−n)z1 − 1 � � ζn(Aj−Aj−n)z2 0 ex fnij � −(x + ζn(Ai − Ai−n)z1)/(σnζn) � dx = fnij(0) � eζn(Ai−Ai−n)z1 − 1 � � eζn(Aj−Aj−n)z2 − 1 � + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)z1z2| (|z1| + |z2|)2 eζn(|Ai−Ai−n||z1|+|Aj−Aj−n||z2|)� , uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Treating in a similar manner the second term, we conclude that the expression inside the square brackets in the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) equals fnij(0) � eζn(Ai−Ai−n)z1 − 1 �� eζn(Aj−Aj−n)z2 − 1 � + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| (1 + |z1|3)(1 + |z2|3) eζn|(Ai−Ai−n)z1|+ζn|(Aj−Aj−n)z2|� , uniformly for i, j ≥ 0 with i ̸= j and z1, z2 ∈ R, and substitution into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='87) gives us E (Zn−i−1Zn−j−11(E0)) = cnij � fnij(0) � ∞ −∞ z1eζn(Ai−Ai−n)z1FZ(dz1) � ∞ −∞ z2eζn(Aj−Aj−n)z2FZ(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| �� Chakrabarty and Samorodnitsky/Clustering of large deviations 26 = Kn(2π)1/2fnij(0) � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='88) + cnijo � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � , as n → ∞, uniformly for i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Recalling that EZ = 0, we see that � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) = O (ζn(Ai − Ai−n)) , and likewise for the second integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since Kn = O(cnij), the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma is an important step in the proof of the main result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the previous lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7 are needed for this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We denote Yni = Zn−i−1 − � 1 + ζ−2 n σ−2 n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz), i ∈ Z, n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='89) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then sup n≥1,i≥0 E � Y 2 ni ��E0 � < ∞ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) and E � YniYnj ��E0 � = −σ−2 n σ4 Z (Ai − Ai−n) (Aj − Aj−n) (1 + o(1)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) as n → ∞, uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) is similar (and much easier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We write P(E0) = Kn(2π)1/2 � ∞ 0 e−xfn � x/(ζnσn) � dx, with Kn as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='78) and simple integration, P(E0) =Kn(2π)1/2 � o � ζ−2 n σ−2 n � + (2π)−1/2 � ∞ 0 exp � −x − x2/(2ζ2 nσ2 n) � dx � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92) =Kn � 1 − ζ−2 n σ−2 n (1 + o(1)) � , n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In combination with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='85) this means that E (Zn−i−1Zn−j−11(E0)) P(E0) = K2 n � � 1 − ζ−2 n σ−2 n � � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � σ−2 n |(Ai − Ai−n)(Aj − Aj−n)| � � , n → ∞, Chakrabarty and Samorodnitsky/Clustering of large deviations 27 uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84), E (Zn−i−11(E0)) E (Zn−j−11(E0)) = K2 n � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � K2 nσ−2 n |Ai − Ai−n||Aj − Aj−n| � , we conclude that E (Zn−i−1Zn−j−11(E0)) P(E0) − E (Zn−i−11(E0)) E (Zn−j−11(E0)) = −K2 nζ−2 n σ−2 n � ∞ −∞ z1 Gζn(Ai−Ai−n)(dz1) � ∞ −∞ z2 Gζn(Aj−Aj−n)(dz2) + o � K2 nσ−2 n |Ai − Ai−n||Aj − Aj−n| � = −K2 nσ−2 n σ4 Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) as n → ∞, uniformly in i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Dividing both sides by P(E0)2 and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92), we obtain E �� Zn−i−1 − E(Zn−i−1|E0) �� Zn−j−1 − E(Zn−j−1|E0) ����E0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='93) = −σ−2 n σ4 Z(Ai − Ai−n)(Aj − Aj−n) (1 + o(1)) , as n → ∞, again uniformly for i, j ≥ 0 with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='92) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='84) E (Zn−i−1|E0) = � 1 + ζ−2 n σ−2 n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz) + o � ζ−1 n σ−2 n |Ai − Ai−n| � , with a similar statement for Zn−j−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='93) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We proceed with establishing conditional distributional limits of certain trun- cated sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For 0 < δ < L denote Sn(j, δ, L) = [nβL]−1 � i=[nβδ] (Ai+j − Ai)Yni + n−1 � i=n−j (Ai+j − Ai+j−n − Ai)Yni (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='94) + n+[nβL] � i=n (Ai+j − Ai+j−n − Ai + Ai−n)Yni, n ≥ 1, j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' With the overshoot T ∗ n as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='49), we have, conditionally on E0, � ζnT ∗ n, � n2α−2Sn([nβt], δ, L), t ≥ 0 �� ⇒ � T0, � (1 − α)−1σZ �� L δ � (s + t)1−α − s1−α� dB1(s) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='95) + � t 0 (t − s)1−αdB2(s) + � L 0 � s1−α − (s + t)1−α� dB3(s) � , t ≥ 0 �� Chakrabarty and Samorodnitsky/Clustering of large deviations 28 in finite dimensional distributions as n → ∞, where T0 is a standard exponen- tial random variable independent of independent standard Brownian motions B1, B2, B3, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For n ≥ 1 and t ≥ 0 we write ξ1◦ n (t) = [nβt] � i=1 AiYni, ξ2◦ n (t) = n−1 � i=n−[nβt] AiYni, ξ3◦ n (t) = n+[nβt] � i=n+1 (Ai − Ai−n) Yni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='4 that, conditionally on E0, � ζnT ∗ n, � n2α−2ξ1◦ n (t) : t ≥ 0 � , � nα−β/2−1ξ2◦ n (t) : t ≥ 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96) � nα−β/2−1ξ3◦ n (t) : t ≥ 0 �� ⇒ � T0, � K1/2 1 σZB1(t3−2α) : t ≥ 0 � , � (1 − α)−1σZB2(t) : t ≥ 0 � , � (1 − α)−1σZB3(t) : t ≥ 0 �� because the difference between the two processes vanishes in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' For ex- ample, n2α−2ζ−2 n σ−2 n [nβt] � i=1 Ai � ∞ −∞ z GζnAi(dz) = O � n1−2α� = o(1) , and similarly with the other two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, for large n, Sn([nβt], δ, L) = [nβL]−1 � i=[nβδ] (Ai+[nβt] − Ai)Yni + n−1 � i=n−[nβt] (Ai+[nβt] − Ai+[nβt]−n − Ai)Yni + n+[nβL] � i=n (Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n)Yni =: V 1 n (t) + V 2 n (t) + V 3 n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Starting with V 3 n , we write V 3 n (t) = n−(1−α)(1−β) [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='97) Chakrabarty and Samorodnitsky/Clustering of large deviations 29 where for 0 ≤ s ≤ L, fn(s, t) = n(1−α)(1−β) An+[nβs]+[nβt] − A[nβs]+[nβt] − An+[nβs] + A[nβs] An+[nβs] − A[nβs] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It is elementary that for fixed s, t, as n → ∞, An+[nβs]+[nβt] − An+[nβs] ≪ A[nβs]+[nβt] − A[nβs] ∼ (1 − α)−1nβ(1−α) � (s + t)1−α − s1−α� , while An+[nβs] − A[nβs] ∼ (1 − α)−1n1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, lim n→∞ fn(s, t) = s1−α − (s + t)1−α =: f(s, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='98) and the limit is easily seen to be uniform in 0 ≤ s ≤ L and t in a compact interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We will show that, conditionally on E0, � n2α−2V 3 n (t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99) ⇒ � σZ(1 − α)−1 � L 0 � s1−α − (s + t)1−α� dB3(s), t ≥ 0 � in finite-dimensional distributions, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' To this end, set cnj(k, t) = inf (j−1)L/k≤s≤jL/k fn(s, t), k ≥ 1, 1 ≤ j ≤ k , and eni(k, t) = fn � n−βi, t � − cn,⌈L−1n−βki⌉(k, t) ≥ 0, k ≥ 1, 1 ≤ i ≤ [nβL] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='98) and monotonicity, lim n→∞ cnj(k, t) = f � (j − 1)k−1L, t � , 1 ≤ j ≤ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='100) A standard continuity argument shows that lim k→∞ lim sup n→∞ sup t∈A max 1≤i≤[nβL] eni(k, t) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='101) for any compact set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have [nβL] � i=1 cn,⌈L−1n−βki⌉(k, t)(An+i − Ai)Yn,n+i = k′ � j=1 cnj(k, t) � i∈ � k−1Lnβ(j−1),k−1Lnβj � ∩Z (An+i − Ai)Yn,n+i = k′ � j=1 cnj(k, t) � ξ3◦ n � k−1Lj � − ξ3◦ n � k−1L(j − 1) �� =: Wnk(t) , Chakrabarty and Samorodnitsky/Clustering of large deviations 30 where k′ = ⌈L−1n−βk[nβL]⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='100), implies that for fixed k, as n → ∞, � nα−β/2−1Wnk(t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='102) ⇒ � (1 − α)−1σZ k � j=1 f � (j − 1)k−1L, t � � B3(k−1jL) − B3(k−1(j − 1)L) � , t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We have [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i − Wnk(t) = [nβL] � i=1 eni(k, t) (An+i − Ai) Yn,n+i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) that, for large n, sup i,j≥0:i̸=j (Ai − Ai−n) (Aj − Aj−n) E (YniYnj|E0) ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' This, along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='90) and the non-negativity of each eni, implies that for large n, E \uf8eb \uf8ec \uf8ed \uf8ee \uf8f0 [nβL] � i=1 eni(k, t) (An+i − Ai) Yn,n+i \uf8f9 \uf8fb 2�����E0 \uf8f6 \uf8f7 \uf8f8 ≤ [nβL] � i=1 [eni(k, t) (An+i − Ai)]2 E(Y 2 n,n+i|E0) = O \uf8eb \uf8ed max 1≤j≤[nβL] enj(k, t)2 [nβL] � i=1 (An+i − Ai)2 \uf8f6 \uf8f8 = O � n2−2α+β max 1≤j≤[nβL] enj(k, t)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Invoking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='101) we conclude that for any compact set A, lim k→∞ lim sup n→∞ n2α−β−2 sup t∈A E �� Wnk(t) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='103) − [nβL] � i=1 fn � n−βi, t � (An+i − Ai) Yn,n+i �2�����E0 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 31 As k → ∞, the process in the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='102) converges in finite- dimensional distributions to the process in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since (2α− 2)− (1 − α)(1 − β) = α− β/2 − 1, the claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='97) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='103) by the “convergence together” argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2 in Billingsley (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' A nearly identical argument shows that, conditionally on E0, � n2α−2V 2 n (t), t ≥ 0 � ⇒ � −σZ(1 − α)−1 � t 0 (t − s)1−αdB2(s), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='104) d= � σZ(1 − α)−1 � t 0 (t − s)1−αdB2(s), t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The situation with the term V 1 n is, once again, similar, with a small twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since lim n→∞ A[nβs]+[nβt] − A[nβs] A[nβs] = (s + t)1−α − s1−α s1−α uniformly for δ ≤ s ≤ L and t, our argument now shows that, conditionally on E0, � n−(2−2α)V 1 n , t ≥ 0 � ⇒ � σZK1/2 1 � L δ (s + t)1−α − s1−α s1−α M(ds), t ≥ 0 � in finite-dimensional distributions, where M is a centred Gaussian random mea- sure with the variance measure with the density (3 − 2α)s2−2α, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since the centred Gaussian random measures (1 − α)−1B3(ds) and K1/2 1 M(ds)/s1−α have the same variance measure, this means that, conditionally on E0, � n2α−2V 2 n (t), t ≥ 0 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='105) ⇒ � σZ(1 − α)−1 � L δ � (s + t)1−α − s1−α� dB3(s), t ≥ 0 � in finite-dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='99), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='104) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='105) are all consequences of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='96), the conver- gence statements they contain hold jointly, and jointly with ζnT ∗ n ⇒ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='95) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The next lemma treats the sequence of shifts appearing due to conditioning on E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Define µn(t) = n2α−2 ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � � ∞ −∞ z Gζn(Ai−Ai−n)(dz), for t ≥ 0 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Then µn → µ∞ as n → ∞, in D([0, ∞)) equipped with the Skorohod J1 topology, where µ∞(t) = −εt3−2α, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Chakrabarty and Samorodnitsky/Clustering of large deviations 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Writing µn(t) =n2α−2ζn ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � � Ai − Ai−n � +n2α−2 ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n � �� ∞ −∞ z Gζn(Ai−Ai−n)(dz) − ζn � Ai − Ai−n �� =: µ(1) n (t) + µ(2) n (t), t ≥ 0, the claim of the lemma will follow once we prove that µ(1) n → µ∞ in D([0, ∞)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106) and µ(2) n (t) → 0 uniformly on compact intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107) We start by proving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Fix L > 0 so that 0 ≤ t ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose first that 1/2 < α < 5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) ��µ(2) n (t) �� =O � n2α−2ζ2 n ∞ � i=0 ��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �� � Ai − Ai−n �2 � =O � n2α−2ζ2 nnβ ∞ � i=1 i−α� Ai − Ai−n �2 � = O � n2α−2ζ2 nnβn3−3α� → 0 uniformly in 0 ≤ t ≤ L, showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' On the other hand, if α ≥ 5/6, then κ ≥ 3 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5), so by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='19) ��µ(2) n (t) �� =O � n2α−2ζ3 n ∞ � i=0 ��Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �� � Ai − Ai−n �3 � =O � n2α−2ζ3 nnβ ∞ � i=1 i−α� Ai − Ai−n �3 � = O � n2α−2ζ3 nnβn4−4α� → 0 uniformly in 0 ≤ t ≤ L, again showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We now prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The pointwise convergence is clear: for fixed t, µ(1) n (t) = σ2 Zσ−2 n n2α−1ε ∞ � i=0 � Ai+[nβt] − Ai+[nβt]−n � (Ai − Ai−n) − n2α−1ε → −εt3−2α Chakrabarty and Samorodnitsky/Clustering of large deviations 33 as n → ∞, where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Next, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='28) we can write for t ≥ 0, µ(1) n (t) =n2α−2ζn 2 �n−1 � i=0 � Ai − Ai−[nβt] �2 + ∞ � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 � =: µ(11) n (t) + µ(12) n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The claim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='106) will follow once we show that both µ(11) n and µ(12) n converge in D([0, ∞)) to continuous limits (both constant factors of µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The fact that µ(11) n converges pointwise to a constant factor of of the pointwise limit of µ(1) n is an intermediate step in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since µ(11) n is a monotone function, its convergence in D([0, ∞)) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' We already know that µ(12) n converges pointwise to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let i0 be such that ai is monotone for i ≥ i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Write for t ≥ 0 µ(12) n (t) =n2α−2ζn 2 � ∞ � i=n+i0 � Ai+[nβt] − Ai+[nβt]−n − Ai + Ai−n �2 − n+i0−1 � i=n−[nβt] � Ai+[nβt] − Ai+[nβt]−n − Ai �2 � =: µ(121) n (t) − µ(122) n (t), so it is enough to show that both µ(121) n and µ(122) n converge in D([0, ∞)) to continuous limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Splitting further, we write for t ≥ 0, µ(122) n (t) = n2α−2ζn 2 � n+i0−1 � i=n−[nβt] A2 i+[nβt]−n + n+i0−1 � i=n−[nβt] � Ai − Ai+[nβt] �� Ai − Ai+[nβt] − 2Ai+[nβt]−n � � =: µ(1221) n (t) + µ(1222) n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Clearly, µ(1221) n (t) = n2α−2ζn 2 [nβt]+i0−1 � i=0 A2 i converges pointwise to a constant factor of µ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Since µ(1221) n is monotone, we conclude that µ(1221) n converges in D([0, ∞)) to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' In order to Chakrabarty and Samorodnitsky/Clustering of large deviations 34 prove that so does µ(122) n , we will show that µ(1222) n (t) → 0 uniformly on compact intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Considering once again 0 ≤ t ≤ L, we have ��µ(1222) n (t) �� ≤ n2α−2ζn 2 n+i0−1 � i=n−[nβt] � Ai+[nβt] − Ai ��� Ai+[nβt] − Ai � + 2Ai+[nβt]−n � = O \uf8eb \uf8edn2α−2ζn n+i0−1 � i=n−[nβt] nβn−α� nβn−α + nβ(1−α)� \uf8f6 \uf8f8 = O � nα−2ζnn3β−βα� → 0 uniformly over 0 ≤ t ≤ L, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Finally, we already know that µ(121) n converges pointwise to a continuous limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Furthermore, by the choice of i0, µ(121) n is a monotone function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Therefore, it converges in D([0, ∞)), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' The following is the final lemma before we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Suppose that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='5) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Let Sn(j) = j+n−1 � i=j Xi, j ≥ 0, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='108) As n → ∞, conditionally on E0, � n−(2−2α) � Sn([nβt]) − nε � , t ≥ 0 � ⇒ � (2Cα)1/2BH(t) + ε−1Cασ2 ZT0 − εt3−2α, t ≥ 0 � in finite-dimensional distributions, where (BH(t) : t ≥ 0) is the standard frac- tional Brownian motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='8) with the Hurst exponent H given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='7), Cα is the constant defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='10), and T0 is a standard exponential random variable independent of the fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQf-f7W/content/2301.01936v1.pdf'} +page_content='91) and the eventual monotonicity of the sequence (An) that there is i0 ≥ 0 such that for all large n, sup i0≤i