diff --git "a/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/load_file.txt" "b/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/CtAzT4oBgHgl3EQfiP1R/content/tmp_files/load_file.txt" @@ -0,0 +1,662 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf,len=661 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='01496v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='DS] 4 Jan 2023 UNIQUE ERGODICITY OF SIMPLE SYMMETRIC RANDOM WALKS ON THE CIRCLE KLAUDIUSZ CZUDEK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Fix an irrational number α and a smooth, positive, real function p on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If current position is x ∈ R/Z then in the next step jump to x + α with probability p(x) or to x − α with probability 1 − p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In 1999 Sinai has proven that if p is asymmetric (in certain sense) or α is Diophantine then the Markov process possesses a unique stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Next year Conze and Guivarc’h showed the uniqueness of stationary distribution for an arbitrary irrational angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In this note we present a new proof of latter result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Introduction Fix an irrational number α ∈ R, and consider the family of Markov processes with the evolution governed by the transition kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) p(x, ·) = p(x)δx+α + q(x)δx−α, p : T × B(T) → [0, 1], where B(S1) stands for the σ-algebra of Borel subsets of S1 and q(x) = 1 − p(x), x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We call the function p symmetric if � T f(x)dx = 0, where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) f(x) = ln p(x) q(x), x ∈ T, and asymmetric otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We call a measure µ invariant for transition kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) if distributing the starting point according to µ makes the Markov process with this transition kernel stationary (thus µ is called also often a stationary measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since T is compact, the Krylov-Bogoliubov technique yields existence of an invariant distribution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) for every choice of continuous p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' However, it is far from being obvious if there exists more than one invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The earliest paper known to the author dealing with similar (but still slightly different) system was by Sine [Sin79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' More recently it was proven by Sinai in [Sin99] that if p ∈ C∞(T) is asymmetric or p ∈ C∞(T) is symmetric and α is Diophantine then the uniqueness follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' One year later Conze and Guivarc’h proved in [CG00] that in the symmetric case p(x) q(x+α) ∈ BV implies uniqueness no matter if α is Diophantine or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The present paper contains another proof of the latter statement assuming p ∈ C1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The advantage of the new proof is that it gives more insight to the problem of mixing and the problem of uniqueness 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Primary 37A50, 60F05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' random rotations, Diophantine approximation, random walk, circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 1 2 KLAUDIUSZ CZUDEK in higher dimensional analogs (where T is replaced by Td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' See Section 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The strategy is based on Sinai’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' To explain it, fix x ∈ T and consider a Markov process (Xn) started at x with transition kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It is evident that the process can achieve only the points of the form x+jα, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus to learn the distribution of (Xn) on T we consider a Markov chain (ξn) on Z, started at 0, with P(ξn+1 = k + 1|ξn = k) = p(x + kα) and P(ξn+1 = k − 1|ξn = k) = q(x + kα) for n ≥ 0 and k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us now restrict to the symmetric case, which is in our scope of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In that case the system on Z is recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p ∈ C∞(T) is symmetric and α is Diophantine then the cohomological equation f(x) = g(x + α) − g(x), where f is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2), possesses a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Using the solution g we can easily check that the measure with density h(z)/q(z) is invariant, where h = exp(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Now the whole difficulty in Sinai’s approach was to show the local limit theorem for (ξn) on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' More precisely, in the symmetric case Sinai has proven that P(ξn = k) ∼ h(x + kα) p(x + kα) 1 √ 2πσ2n exp −k2 2nσ2 , for some σ > 0 and all x ∈ T, where ∼ means the ratio of both sides tends to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' With this fact one can show that Eϕ(Xn) → � T ϕ(z)h(z) q(z) dz, which easily implies the unique ergodicity (in fact it’s even a stronger property called mixing or stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Unfortunately we cannot follow exactly the same path when generalizing result to all irrational α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Recently Dolgopyat, Fayad and Saprykina [DFS21] have proven that if α is Liouville then the behaviour of (ξn) on Z is erratic for the generic choice of smooth and symmetric p (see Theorems A-E therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In particular, neither annealed, nor quenched central limit theorem holds (see Corollary D and G therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' However, we can still modify something in Sinai’s idea to get desired assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The main result of this work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p ∈ C1(T) is symmetric and separated from 0 and 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 0 < p(x) < 1 for each x ∈ T) then there exists exactly one invariant measure for the transition kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' As it was mentioned, the proof is some sense is in the spirit of Sinai’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We still concentrate on the process (ξn) on Z but instead of proving the local limit theorem we focus on the limits lim n→∞ P(ξ0 = k) + · · · + P(ξn−1 = k) P(ξ0 = m) + · · · + P(ξn−1 = m), where k, m ∈ Z are two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The problem of existence of such limits for general (including countable space, null recurrent) Markov chains was raised by Kolmogorov in 1936 and answered two years later by Doeblin [Doe38] without identification of the value of the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It has been done only later by Chung [Chu50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It turns out we can define certain infinite measure on Z, k �−→ ax,k (depending on x ∈ T since UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 3 (ξn) depends on x ∈ T) such that the limit above tends to ax,k/ax,m for arbitrary two states k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In Section 2 we identify the measure k �−→ ax,k on Z and reproduce the proof of Doeblin ratio limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In Section 3 it is proved that if one takes a large interval of integers A of length q and projects the measure k �−→ ax,k to the circle (by identifying k with x + kα) then what we obtain is, after normalization and up to ε, independent of the choice of the interval A and the point x, provided q is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Section 4 contains how to complete the proof of Theorem 1 using the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Section 5 contains some final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Acknowledgments and personal remarks When I proved the main theorem I wasn’t aware of Conze, Guivarc’h result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' After discovering it, I started thinking if my proof can be used to show something more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' I realized the advantage of mine is it can be modified to obtain mixing (assuming p is C1 and symmetric, no matter if α is Diophantine or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Then I gave several talks about it, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' in the conference “Probabilistic techniques in random and time-varying dynamical systems”, Luminy 3-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2022 or in the KTH dynamical systems seminar, where I announced “mixing” result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Although I still think this result is true, I didn’t predicted certain difficulties in the proof and I need more time and effort to complete it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Meanwhile I’m publishing the proof of uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It’s not going to be submitted to any journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The research was supported by the Polish National Science Center grant Pre- ludium UMO-2019/35/N/ST1/02363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Basic facts about symmetric random walks on Z Fix x ∈ T and define (ξn) to be the Markov process on Z, started at 0, with P(ξn+1 = k + 1|ξn = k) = p(x + kα) and P(ξn+1 = k − 1|ξn = k) = q(x + kα) for n ≥ 0 and k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In present section we are going to prove recurrence of this random walk and some related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We say that (ξn) is recurrent if almost surely there exists n > 0 with ξn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We say (ξn) is null recurrent if it is recurrent and the expected time of the first return to 0 is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p is of bounded variation, symmetric and separated from 0 and 1 then the process (ξn) is recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Moreover, for every r > 0 there exists m0 that can be chosen uniformly in x ∈ T such that the expected number of returns of (ξn) to zero until m0 is greater than r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' P(ξ1 = 0) + · · · + P(ξn = 0) = E � 1{0}(ξ1) + · · · + 1{0}(ξn) � > r for n ≥ m0, whatever x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' To show the recurrence of (ξn), we reproduce the analysis from [DFS21], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us define a function M : Z → R by M(0) = 0, M(1) = 1, M(n) = 1 + n−1 � k=1 k � j=1 q(x + jα) p(x + jα) for n ≥ 2, 4 KLAUDIUSZ CZUDEK and M(−n) = − n � k=0 k � j=0 p(x − jα) q(x − jα) for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' To avoid complicated notation, we do not stress the dependence of M on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It can be checked that (M(ξn)) is a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let a < 0 < b and let us define τ to be the first moment when (ξn) hits a or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By Doob’s theorem EM(ξτ) = M(ξ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' On the other hand EM(ξτ) = M(a)P(ξτ = a) + M(b)P(ξτ = b) = M(a)P(ξτ = a) + M(b)(1 − P(ξτ = a)), which combined with EM(ξτ) = 0 yields P(ξτ = a) = M(b) M(b) − M(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If ξτ = a then (ξn) returns to 0 before hitting b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Setting a = −1 above we get therefore (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) P � (ξn) returns to 0 before hitting b � ≥ M(b) M(b) − M(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Similarly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) P � (ξn) returns to 0 before hitting a � ≥ −M(a) M(1) − M(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' This easy implies the random walk (ξn) is recurrent provided M(n) → ∞ as n → ∞ and M(n) → −∞ as n → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The latter is implied by the following consequence of the Denjoy-Koksma inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' For every A > 0 there exists n0 > 0 that is independent of x ∈ T such that M(n) > A for n ≥ n0 and M(n) < −A for n ≤ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The function M(n) is a sum of expressions of the form k � j=1 q(x + jα) p(x + jα) for k < n, therefore to show the assertion it is sufficient to find δ > 0 such that the product above is greater than δ for infinitely many k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Define f(x) = ln q(x) − ln p(x), x ∈ T, and observe we can write k � j=1 q(x + jα) p(x + jα) = exp � k � j=1 f(x + jα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The function f is of bounded variation and � T f(t)dt = 0 so the Denjoy-Koksma inequality (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1 in [Her79], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 73) yields ���� q � j=1 f(x + jα) ���� = ���� q � j=1 f(x + jα) − q � T f(t)dt ���� < var(f) UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 5 for an arbitrary x ∈ T and an arbitrary closest return time q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' But this means for an arbitrary closest return time q we have exp � k � j=1 f(x + jα) � > e−var(f) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus the assertion follows with δ = e−var(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ To show the remaining part of Proposition, fix r > 0 and take ε > 0 so small that (1 − ε)2r > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) and Lemma 1 there exists a > 0 (suitable for all x ∈ T) such that P � (ξn) returns to 0 before hitting −a or a � ≥ 1 − ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since p and q are separated from 0, there exists n0 so large (suitable for all x ∈ T) such that probability that (ξn) stays in (−a, a) for the first n0 steps is less than ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Combining these two facts yields P � (ξn) returns to 0 before n0 � > 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By the strong Markov property P � (ξn) returns 2r-times to 0 before 2rn0 � > (1 − ε)2r > 1/2, by the choice of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The assertion follows with m0 = 2rn0 since the expected number of returns to 0 before m0 is greater than 2r with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ It is advantageous to use the following notation in the remaining part of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let pn i,j denote the probability of transition from state i to state j in n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We simply write pi,j instead of p1 i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let kpn i,j stands for the probability of transition from state i to state j in n steps under the restriction that state k is visited in neither of steps 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Again, these values depend on chosen point x ∈ T but we refrain from stressing that in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Clearly jpn k,j is the probability of the first visit in j starting at k occurring in step n and kpn k,j is the probability of transition to j from k in n steps with the restriction that the state k is not visited in steps 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The series �∞ n=1 kpn k,j is interpreted as the expected number of visits in j starting at k before the first return to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It is not difficult to show the convergence of this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p is of bounded variation, symmetric and separated from 0 and 1 then the series �∞ n=1 kpn i,j is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Moreover, for any q ≥ 1 its sum is uniformly bounded over all k, i, j with |k − i|, |k − j|, |j − i| < q, x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' For every ε > 0 and natural q ≥ 1 there exists N with �∞ n=N kpn i,j < ε whatever x ∈ T, provided |k − j| ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let m ∈ N be such that kpm j,k > η for some η > 0 and all j, k with the same parity and |j − k| ≤ q (remember the Markov chain is periodic with period two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It is clear m and η can be chosen uniformly in x ∈ T since p is separated from 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We have kpn i,j · kpm j,k ≤ kpn+m i,k 6 KLAUDIUSZ CZUDEK for n ∈ N, hence ∞ � n=N kpn i,j ≤ 1 kpm j,k ∞ � n=N kpn+m i,k ≤ 1 η ∞ � n=N kpn+m i,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The last series represents the probability that the first transition to k starting at i occurs at earliest at the step N + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' This number is bounded from above by ε if N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Moreover, N can be chosen to be suitable for all x ∈ T by a reasoning similar to the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ It is not difficult also to recover the value of �∞ n=1 kpn k,j, which represents the expected value of appearances in state j of the process started at k before it returns to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p is of bounded variation, symmetric and separated from 0 and 1 and ax,n is defined by1 ax,0 = 1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='3) ax,n = q(x) q(x + nα) n−1 � j=0 p(x + jα) q(x + jα) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) ax,−n = p(x) p(x − nα) n−1 � j=0 q(x − jα) p(x − jα) for n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Then ∞ � n=1 kpn k,j = ax,j ax,k for any two states k, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Fix k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' First of all, the aim is to show the assertion for j = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Notice if the process started at k visits k − 1 in the first step then it necessarily visits k before ever reaching k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus the probability of exactly one appearance in k + 1 before returning to k is p(x + kα) · q(x + (k + 1)α) and the probability of exactly r appearances is p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) (since after the first r − 1 visits it “jumps” to the state k + 2 with probability p(x + (k + 1)α) and right after r-th to k with probability q(x + (k + 1)α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Hence the expected number of appearances is ∞ � n=1 kpn k,j = ∞ � r=1 r · p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) = p(x + kα)q(x + (k + 1)α) ∞ � r=1 rp(x + (k + 1)α)r−1 = p(x + kα)q(x + (k + 1)α) (1 − p(x + (k + 1)α))2 = p(x + kα)q(x + (k + 1)α) q(x + (k + 1)α)2 = p(x + kα) q(x + (k + 1)α), where in the passing from the second line to the third one the formula �∞ r=1 rzr−1 = 1 (1−z)2 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since the last equals ax,k+1 ax,k , this completes the proof for j = k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 1In contrast to other symbols here we stress the dependence on x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' That is because this symbol appears in the next section where the dependence on x is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 7 To end the proof we proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us assume the assertion holds for k + 1, k + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=', j for some j > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us consider the process started at k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It is easy to conclude the expected number of appearances of this process in j + 1 under the condition the number of appearances in k + 1 is r equals, by the induction assumption, to r · ax,j+1 ax,k+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In turn, the probability of exactly r visits in k before returning to k is, as before, p(x+kα)·p(x+(k +1)α)r−1 ·q(x+(k +1)α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In the view of foregoing, the expected number of appearances in j + 1 of the process started at k before returning to k equals ∞ � r=1 r · ax,j+1 ax,k+1 p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) = ax,j+1 ax,k+1 p(x + kα) q(x + (k + 1)α) = ax,j+1 ax,k+1 ax,k+1 ax,k = ax,j+1 ax,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' This completes the proof of Lemma 3 in the case of any two integers with j > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The case j < k is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ The last result of this section is basically the Doeblin ratio limit theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Corollary 2 to Theorem 4 in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 48, in [Chu60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' However, reproducing the proof is necessary because we need a kind of uniform convergence result over all x ∈ T and states j, k that are sufficiently close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If p is of bounded variation, symmetric and separated from 0 and 1 then for every ε > 0 and q ≥ 1 there exists N such that ���� P(ξ1 = j) + · · · + P(ξn = j) P(ξ1 = k) + · · · + P(ξn = k) − ax,j ax,k ���� < ε for every n ≥ N, x ∈ T, provided |k|, |j| ≤ q and |k − j| ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By Lemma 2 there exists B > 0 such that �N n=1 kpn 0,j ≤ B for every N and states k, j satisfying the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The number B can be chosen also such that max |j|,|k|≤q max x∈T ax,j ax,k ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Apply Lemma 2 and 3 to get N0 so large that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='5) ���� N−ν � n=1 kpn k,j − ax,j ax,k ���� < ε 3 for N ≥ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The number N ′ 0 > N0 should be so large that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='6) 2B �N n=1 pn 0,k < ε 3 for N ≥ N ′ 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='7) BN0 �N n=1 pn 0,k < ε 3 The easily proven decomposition formula pn 0,j = kpn 0,j + n−1 � ν=1 pν 0,k · kpn−ν k,j 8 KLAUDIUSZ CZUDEK yields N � n=1 pn 0,j = N � n=1 kpn 0,j + N−1 � ν=1 pν 0,k N−ν � n=1 kpn k,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We have ���� P(ξ1 = j) + · · · + P(ξN = j) P(ξ1 = k) + · · · + P(ξN = k) − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� = ���� �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� = ���� �N n=1 kpn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j + �N−1 ν=1 pν 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k �N−ν n=1 kpn k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k − �N n=1 ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� ≤ ���� �N n=1 kpn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k pN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k + �N−1 ν=1 pν 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N−ν n=1 kpn k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� ≤ ���� �N n=1 kpn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k pN 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� + ���� �N−1 ν=N−N0+1 pν 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N−ν n=1 kpn k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� + ���� �N−N0 ν=1 pν 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N−ν n=1 kpn k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j − ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='j ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k � �N n=1 pn 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k ���� By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='5) the third term is less than ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By the very definition of B, the numerator of the first term is less that 2B and the numerator of the second expression is less than BN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='7) complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us consider an interval A ⊆ Z of length q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let (ξn) be as usually the process started at 0, and let τ be the moment of the first visit of (ξn) in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If N is given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since N was independent of x ∈ T, a conditional argument easily implies ������ P(ξ0 = j|Fτ) + · · · + P(ξn−1 = j|Fτ) P(ξ0 = k|Fτ) + · · · + P(ξn−1 = k|Fτ) − ax,j ax,k ���� < ε almost surely on {τ < n − N} for any two states k, j ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us now consider certain function ϕ : Z → R with support contained in an interval A, as above, and ∥ϕ∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' An easy argument using Remark 1 yields ���� E � ϕ(ξ0) + · · · + ϕ(ξn−1) ��Fτ � E � 1A(ξ0) + · · · + 1A(ξn−1) ��Fτ � − � i∈A ϕ(i)ax,i � i∈A ax,i ���� < ε almost surely on {τ < n − N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' It is clear that N can be chosen uniformly over all intervals A of fixed length q, x ∈ T and function ϕ as far as ∥ϕ∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Projection of measures Put ax,k = exp � Φ(x) + · · · + Φ(x + (k − 1)α) �1 + exp Φ(x + kα) 1 + exp Φ(x) for k ≥ 1 and ax,0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Define µx,n = 1 Mx,n n−1 � k=0 ax,kδx+kα UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 9 for x ∈ T and n ≥ 1, where Mx,n is the normalizing constant, Mx,n = n−1 � k=0 ax,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If x ∈ T, k1, k2 ∈ N, then ax,k1+k2 = ax,k1 · ax+k1α,k2 and µx,k1+k2 = Mx,k1 Mx,k1+k2 µx,k1 + ax,k1 Mx+k1α,k2 Mx,k1+k2 µx+k1α,k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The proof is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' For every ε > 0 there exists N such that if q ≥ N is the closest return time then (1 − ε)ay,n ≤ ax,n ≤ (1 + ε)ay,n for every natural n ≤ q and x, y ∈ T with |x − y| < 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We can find n0 so large that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) 1 n � f ′� x � + · · · + f ′� x + (n − 1)α �� < δ for n ≥ n0 and every x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Indeed, this is the consequence of the Birkhoff ergodic theorem applied to the rotation by angle α and the Lebesgue measure (uniform convergence in x follows from unique ergodicity and continuity of f ′, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='13 in [KH95]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let q ≥ n0 be so large that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) 1 q � f ′� x � + · · · + f ′� x + jα �� < δ for j ≤ n0 and every x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Finally, by uniform continuity, let us assume q to be so large that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='3) 1 − δ ≤ 1 + exp f(x) 1 + exp f(y) ≤ 1 + δ for x, y ∈ T, |x − y| ≤ 2/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take x, y ∈ T with |x − y| ≤ 2/q, a natural n ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' By the mean value theorem there exists z in the shorter arc joining x and y such that ax,n ay,n = exp �� f ′(z) + · · · + f ′(z + (n − 1)α) � |x − y| � ×1 + exp f(x) 1 + exp f(y) · 1 + exp f(x + (n + 1)α) 1 + exp f(y + (n + 1)α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If n ≥ n0 then apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) and the fact that |x − y| ≤ 2/q to get � f ′(z)+· · ·+f ′(z +(n−1)α) � |x−y| ≤ 1 n � f ′� x � +· · ·+f ′� x+(n−1)α �� n q ≤ 2δ, as n ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' This combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='3) yields e−2δ(1 − δ)2 ≤ ax,n ay,n ≤ e2δ(1 + δ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='3) we can deduce similar statement in the case n < n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If δ → 0 then the values on the left and right above tend to 1, thus the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ 10 KLAUDIUSZ CZUDEK Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let ϕ ∈ C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' For every ε > 0 there exists N such that if q ≥ N is a closest return time then���� � T ϕdµx,q − � T ϕdµy,q ���� < ε for every x, y ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take η > 0 and ϕ ∈ C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Choose δ > 0 small (to be determined), and let q be the closes return time such that Lemma 5 is satisfied with ε replaced by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' As a consequence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) 1 − δ < az1,n az2,n < 1 + δ and 1 − δ < Mz1,n Mz2,n < 1 + δ for n ≤ q and z1, z2 ∈ T with |z1 − z2| < 2/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Further, using Lemma ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' we easily see az,qn → 1 uniformly in z, when (qn) is the sequence of closest return times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus q can be chosen so large that 1 − δ ≤ az,q ≤ 1 + δ for all z ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Using the first assertion in Lemma 4 it implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='5) 1 − δ ≤ az,naz+nα,n−q ≤ 1 + δ for n < q and z ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The last thing we want to assume on q it is so large that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='6) sup z∈T sup |h|≤ 2 q |ϕ(z + h) − ϕ(z)| < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us take x, y ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Denote xj = x + jα, yj = y + jα, j ∈ [0, q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let t be the smallest natural number with d(xt, y) ≤ 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since rotation is an isometry we immediately see d(xt+j, yj) < 1 q for j = 0, 1, · · · q − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' In particular d(xq, yq−t) < 1 q , hence d(yq−t, x) ≤ d(yq−t, xq) + d(xq, x) < 1/q + 1/q = 2/q and, since the rotation is isometry, d(yq−t+j, xj) < 2 q for j = 0, · · · , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The measure µx,q is an atomic measure with atoms at the points x, x+α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , x+ (q −1)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The idea is to represent µx,q as a convex combination of measures concen- trated on two disjoint subsets {x, x+α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , x+(t−1)α} and {x+tα, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , x+(q−1)α} and, similarly, represent µy,q and a convex combinations of measures concentrated on two disjoint subsets {y, y +α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=', y +(q −t−1)α} and {y +(q −t)α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' , y +qα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Namely, it is easy to check using Lemma 4 that µx,q = Mx,t Mx,q µx,t + ax,t Mxt,q−t Mx,q µxt,q−t and µy,q = My,q−t My,q µy,q−t + ay,q−t Myq−t,t My,q µyq−t,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since d(xt, y) ≤ 1/q, in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) we expect the second measure in the decom- position of µx,q to be close to the first measure in decomposition of µy,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Similar reasoning applies to two remaining terms since d(yq−t, x) < 2/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We have ���� � T ϕdµx,q − � T ϕdµy,q ���� ≤ ���� Mx,t Mx,q � T ϕdµx,t − ay,q−t Myq−t,t My,q � T ϕdµyq−t,t ���� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='7) + ����ax,t Mxt,q−t Mx,q � T ϕdµxt,q−t − My,q−t My,q � T ϕdµy,q−t ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 11 Let us now focus on the second term on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' The analysis of the first term proceeds analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We have ����ax,t Mxt,q−t Mx,q � T ϕdµxt,q−t − My,q−t My,q � T ϕdµy,q−t ���� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8) ≤ ����ax,t Mxt,q−t Mx,q − My,q−t My,q ���� � T |ϕ|dµxt,q−t +My,q−t My,q ���� � T ϕdµxt,q−t − � T ϕdµy,q−t ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We are going to show the first term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8) is bounded by ∥ϕ∥∞η and the second by δ + ∥ϕ∥∞η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since exactly the same estimates can be derived for the first term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='7), it will give ���� � T ϕdµx,q − � T ϕdµy,q ���� ≤ 2δ + 4∥ϕ∥∞η and will complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus what remains to do is to find the desired bounds on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Analysis of the first term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8) We have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='9) ����ax,t Mxt,q−t Mx,q − My,q−t My,q ���� = My,q−t My,q ����ax,t · My,q Mx,q Mxt,q−t My,q−t − 1 ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since d(y, xt) < 1/q ≤ 2/q we can apply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) to get that 1 − δ ≤ Mxt,q−t My,q−t ≤ 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Further, d(yq−t, x) ≤ 2/q, thus Lemma 4 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) give My,q = My,q−t + ay,q−tMyq−t,t ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='5) we have axt,q−t ≤ 1+δ ax,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Finally My,q ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t ≤ (1 + δ)Mxt,q−t + (1 + δ)3 ax,t Mx,t ≤ (1 + δ)3 � Mxt,q−t + 1 ax,t Mx,t � = (1 + δ)3 ax,t � ax,tMxt,q−t + Mx,t � = (1 + δ)3 Mx,q ax,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' So far we used only the bounds from above in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4 ) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Applying the same reasoning with estimates from below we see that My,q ≥ (1 − δ)3 Mx,q ax,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Going back to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='9) we have (1 − δ)4 ≤ ax,t · My,q Mx,q Mxt,q−t My,q−t ≤ (1 + δ)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If δ was chosen sufficiently small then ����ax,t · My,q Mx,q Mxt,q−t My,q−t − 1 ���� < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 12 KLAUDIUSZ CZUDEK Since My,q−t My,q ≤ 1 it leads to the estimate ����ax,t Mxt,q−t Mx,q − My,q−t My,q ���� < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Thus the first term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8) is bounded by η∥ϕ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Analysis of the second term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='8) To deal with the second expression we have clearly My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q ≤ 1 and ���� � T ϕdµxt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t − � T ϕdµy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ���� = ���� q−t−1 � k=0 axt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k Mxt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(xt +kα)− q−t−1 � k=0 ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(y+kα) ���� ≤ ���� q−t−1 � k=0 axt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k Mxt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(xt + kα) − q−t−1 � k=0 ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(xt + kα) ���� + ���� q−t−1 � k=0 ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(xt + kα) − q−t−1 � k=0 ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ϕ(y + kα) ���� ≤ q−t−1 � k=0 axt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k Mxt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ��ϕ(xt + kα) �� ����1 − ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k axt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k Mxt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ���� + q−t−1 � k=0 ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='k My,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='q−t ��ϕ(xt + kα) − ϕ(y + kα) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Since d(xt, y) < 1/q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='4) yields (1 − δ)2 ≤ ay,k axt,k Mxt,q−t My,q−t ≤ (1 + δ)2, thus ����1 − ay,k axt,k Mxt,q−t My,q−t ���� < η if δ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' This leads us to the estimate q−t−1 � k=0 axt,k Mxt,q−t ��ϕ(xt + kα) �� ����1 − ay,k axt,k Mxt,q−t My,q−t ���� ≤ ∥ϕ∥η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Clearly, q−t−1 � k=0 ay,k My,q−t ��ϕ(xt + kα) − ϕ(y + kα) �� ≤ δ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='6), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' □ UNIQUE ERGODICITY OF SIMPLE SYMMETRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Proof of Theorem 1 We shall use the following criterion for the uniqueness of the stationary distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' If for every ε > 0 and nonnegative ϕ ∈ C(T) with 1/2 < ∥ϕ∥∞ < 1 there exist β ∈ R and N > 0 such that ���� ϕ(x) + · · · + P n−1ϕ(x) n − β ���� < ε for every x ∈ T and n ≥ N, then there exists exactly one stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us take ε > 0 and ϕ ∈ C(T) as stated in the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let y ∈ T be arbitrary, and let β = � T ϕdµy,q, where q is chosen so large that Proposition 3 holds with ε replaced by ε/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Take x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Set Ak = [kq, (k + 1)q), k ∈ Z, and define ϕk(j) = 1Ak(j) · ϕ(x + jα), ϕk : Z → R, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Observe that � i∈Ak ϕk(i)ax,i � i∈Ak ax,i = � T ϕdµx+kα,q for every k, thus Proposition 3 gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) ���� � i∈Ak ϕk(i)ax,i � i∈Ak ax,i − β ���� < ε 3, for an arbitrary k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' For k ∈ Z denote by τk the moment of the first visit of (ξn) in Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Fix n sufficiently large and set Γ ⊆ Z to be the set of k’s such that Ak is visited with positive probability till n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Apply Proposition 2 and Remark 2 to get a number N such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='2) ���� E � ϕk(ξ0) + · · · + ϕk(ξn−1) ��Fτk � E � 1Ak(ξ0) + · · · + 1Ak(ξn−1) ��Fτk � − β ���� < ε a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' on {τk < n − N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let (Xn) be the process with transition kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1) started at x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' We have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='3) |E � ϕ(X0) + · · · + ϕ(Xn) � − βn| = ����E � � k∈Γ ϕk(ξ0) + · · · + ϕk(ξn−1) � − βE � � k∈Γ 1Ak(ξ0) + · · · + 1Ak(ξn−1) ����� ≤ � k∈Γ E ����E � ϕk(ξ0) + · · · + ϕk(ξn−1) ��Fτk � − βE � 1Ak(ξ0) + · · · + 1Ak(ξn−1) ��Fτk ����� = � k∈Γ E � E � 1Ak(ξ0) + · · · + 1Ak(ξn−1) ��Fτk � ���� E � ϕk(ξ0) + · · · + ϕk(ξn−1) ��Fτk � E � 1Ak(ξ0) + · · · + 1Ak(ξn−1) ��Fτk � − β ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' Let us fix k ∈ Γ and split the expectation above as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1Ak(ξ0) + · · · + 1Ak(ξn−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='��Fτk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='ϕk(ξ0) + · · · + ϕk(ξn−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='��Fτk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='1Ak(ξ0) + · · · + 1Ak(ξn−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='��Fτk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� − β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='KLAUDIUSZ CZUDEK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAzT4oBgHgl3EQfiP1R/content/2301.01496v1.pdf'} +page_content='= E1{τk