*

#  Weak Convergence for Self-Normalized Partial Sum Processes in the Skorokhod
$M_{1}$ Topology with Applications to Regularly Varying Time Series

Christis Katsouris
University of Southampton School of Economic, Social and Political Sciences,
University of Southampton, Southampton SO17 1BJ, UK. Email:
<EMAIL_ADDRESS>
###### Abstract

In this paper we study the weak convergence of self-normalized partial sum
processes in the Skorokhod $M_{1}$ topology for sequences of random variables
which exhibit clustering of large values of the same sign. We show that for
stationary regularly varying sequences with such properties, their
corresponding properly centered self-normalized partial sums processes
converge to a stable Lévy process. The convergence is established in the space
of cádlág functions endowed with Skorohod’s $M_{1}$ topology, which is more
suitable especially for cases in which the standard $J_{1}$ topology fails to
induce weak convergence of joint stochastic functionals.

###### Contents

  1. 1 Introduction
    1. 1.1 Preliminary Theory
  2. 2 Stationary regularly varying sequences
    1. 2.1 Regular variation
    2. 2.2 Point processes and dependence conditions
    3. 2.3 The weak and strong $M_{1}$ topologies
  3. 3 Joint Functional Convergence of Partial Sum Processes
    1. 3.1 Joint Functional Convergence of Partial Sum Processes with $\alpha\in(0,1)$
    2. 3.2 Joint Functional Convergence of Partial Sum Processes with $\alpha\in[1,2)$
    3. 3.3 Weak Convergence of Self-Normalized Partial Sum Processes
  4. 4 Main Applications
  5. 5 Conclusion
  6. A Proofs of Main Results in Section 1
    1. A.1 Proof of Lemma 1
    2. A.2 Proof of Lemma 3
  7. B Proofs of Main Results in Section 2
    1. B.1 Proof of Lemma 5
    2. B.2 Proof of Theorem 1
    3. B.3 Proof of Theorem 2
    4. B.4 Proof of Theorem 3
    5. B.5 Proof of Theorem 4

##  Introduction

Weak convergence theory in the space of bounded functions (see, Kuelbs,
(1968), Andersen and Dobric, (1987) Hoffmann-Jorgensen, (1991), Arcones,
(1996) and Peterson, (2011)) is commonly used both in statistical as well as
in econometric environments with time series regressions111Notice that time
series sequences are stochastic processes models which are classified into two
types: one is for continuous type processes such as diffusion processes and
others are for jump processes such as point processes.; especially when
asymptotic theory of model estimators and associated test statistics has to be
developed. Moreover, the literature on the weak convergence of self-normalized
processes (see, Csorgo et al., (2003)) - that belong to the class of diffusion
processes based on the standard $J_{1}$ topology, covers joint functional
convergence of functionals for both stationary (see, Kulperger and Yu, (2005))
as well as nonstationary stochastic processes (see, Buchmann and Chan,
(2007)). However, there are cases in which weak convergence fails to hold
(see, Bucher et al., (2014)), such as empirical copula and tail dependence
processes that have corresponding limit processes with discontinuous
trajectories. In this study, we establish the weak invariance principle for
self-normalized partial sum processes of stationary sequences based on the
Skorokhod $M_{1}$ topology (see, Basrak et al., (2010), Krizmanić, (2018,
2020, 2022)) which requires some special attention. Therefore, weak
convergence equipped with the $M_{1}$ topology covers cases where the $J_{1}$
topology is not applicable due to discontinuities in trajectories of limit
processes and thus permits to establish related asymptotic theory.

Our fundamental framework for developing the asymptotic theory of self-
normalized partial sum processes of heavy tailed time series is based on the
main results presented in the seminal study of Basrak et al., (2010), BKS, who
establish weak convergence in the $M_{1}$ topology for partial sum process.
Specifically, the authors show that the convergence takes place in the space
of cádlág functions endowed with Skorokhod’s $M_{1}$ topology, while the usual
$J_{1}$ topology is not suitable space to ensure convergence as the partial
sum processes may exhibit rapid successions of jumps within temporal clusters
of large values, collapsing in the limit to a single jump. However, BKS only
consider the case of partial sums and partial maxima sums and there is no
related limit theory for self-normalized partial sum processes of stationary
regularly varying processes. In this paper, we focus on theoretical and
empirical applications for exactly this case.

Various studies in the literature focus on developing asymptotic theory for
self-normalized sums222Weak convergence of self-normalized partial-sum
processes is also employed in the case of nonstationary time series processes
to establish asymptotic approximations. However in this study we only consider
strictly stationary sequences with relevant mixing conditions.  of independent
random variables under certain regularity conditions (see, Shao, (1997), Shao,
(1999) and Fan et al., (2019)) such as a martingale sequence condition on
innovation sequences. More precisely, the asymptotic behaviour of self-
normalized estimators and test statistics which have a martingale
representation can be obtained using principles such as Cramér-type moderate
deviations and Berry-Esseen bounds (see, Bentkus and Gotze, (1996)) and Gao et
al., (2022) who extend to the case of dependent random variables. Fewer
studies exist specifically for self-normalized partial sums of heavy tailed
processes, with a recent approach presented by Matsui et al., (2023), although
in this framework the authors do not consider the condition of small vanishing
values.

###  Preliminary Theory

The development of invariance principles, are based on a well-defined metric
space, such as $d(.,.)$ be the Skorohod metric on $D(0,\infty)$. Since local
uniform convergence implies Skorohod convergence, we get that
$d\big{(}X_{0}^{(\varepsilon)}(\,\cdot\,),X_{0}(\,\cdot\,)\big{)}\to 0$ almost
surely as $\varepsilon\to 0$, and hence, since almost sure convergence implies
weak convergence, $X_{0}^{(\varepsilon)}(\,\cdot\,)\Rightarrow
X_{0}(\,\cdot\,)$.

Consider $\left(X_{i}\right)_{i\geq 1}$ to denote a sequence of independent
random variables with zero means and finite variances. Define with
$S_{n}=\sum_{i=1}^{n}X_{i}$, $B_{n}^{2}=\sum_{i=1}^{n}\mathbb{E}X_{i}^{2}$ and
$V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. Then, the self-normalized sum
$S_{n}/V_{n}\to\mathcal{N}(0,1)$ if and only if the random variable $X$, is in
the domain of attraction of the normal law. In addition, if the self-
normalized sums $S_{n}/V_{n},n\in\mathbb{N}$, are stochastically bounded, then
they are uniformly sub-Gaussian in the sense that

(1.1) $\displaystyle\underset{n\in\mathbb{N}}{\text{sup}}\
\mathbb{E}\left[e^{tS_{n}/V_{n}}\right]\leq 2e^{ct^{2}}\ \ \text{for all}\ \
t\in\mathbb{R}\ \ \text{and some}\ c<\infty.$

The normal law property holds when considering the joint weak convergence of
partial-sum processes. Let $X_{1,n},...,X_{n,n},n\geq 1$, be a triangular
array of independent infinitesimal random variables. Then the joint functional
$(S_{n},V_{n}^{2})$ satisfies the following weak convergence result

(1.2)
$\displaystyle\left(\sum_{i=1}^{n}X_{i,n},\sum_{i=1}^{n}X_{n,n}^{2}\right)\to_{D}(U,V),$

for a nondegenerate pair $(U,V)$.

Moreover, it holds that
$\sum_{i=1}^{n}X_{i,n}\big{/}\sqrt{\sum_{i=1}^{n}X_{i,n}^{2}}\to_{D}\mathcal{N}(0,1)$
iff for some $\kappa>0$,

(1.3)
$\displaystyle\sum_{i=1}^{n}X_{i,n}\overset{d}{\to}\kappa\mathcal{N}(0,1)\ \
\text{and}\ \ \sum_{i=1}^{n}X_{i,n}^{2}\overset{p}{\to}\kappa^{2}.$

An extension of the self-normalized central limit theorem to Donsker type
functional central limit theorem applies. Define the partial-sum process
$S_{[nt]}=\sum_{i=1}^{[nt]}X_{i}$. The concept of invariance principles was
first introduced by the paper of Erdos and Kac, (1946) and Donsker, (1951) as
well as generalized by Prokhorov, (1956) (see, discussion in Kuelbs, (1968)).
Furthermore, Kuelbs, (1968) consider the weak convergence of these random
elements into continuous functionals on the space of real-valued functions
that includes Gaussian measures as analogues to Wiener measures. The
particular property of such topological spaces motivated the construction of a
new metric such as the approach of Skorokhod, (1956), who introduced a number
of metrics on the space of cádlag functions on $[0,1]$, with $J_{1}$ topology
being the most widely used in statistics and econometric asymptotic theory.

Suppose we consider the asymptotic behaviour of a self-normalized partial-sum
process, as $n\to\infty$ it holds that $S_{[nt]}/V_{n}\to_{D}W(t)$ on
$\left(\mathcal{D}[0,1],\rho\right)$, where $\rho$ is the sup-norm metric for
functions in $\mathcal{D}[0,1]$, and $\left\\{W(t),0\leq t\leq 1\right\\}$ is
a standard Wiener process. The limit result below of Csorgo et al., (2003)
shows that the behaviour in probability of the self-normalized process
$\big{\\{}S_{\lfloor nt\rfloor}/V_{n}\big{\\}}_{t\in[0,1]}$ coincides with the
behaviour of a standard Brownian motion $W=\left\\{W(t)\right\\}_{t\in[0,1]}$.

On an appropriate probability space it holds that

(1.4) $\displaystyle\underset{t\in[0,1]}{\text{sup}}\left|\frac{S_{\lfloor
nt\rfloor}}{V_{n}}-\frac{W(nt)}{\sqrt{n}}\right|=o_{p}(1).$

In this article, we consider the asymptotics for self-normalized partial-sum
processes where joint weak convergence arguments of functionals to the $J_{1}$
topology do not directly apply. The reason is that this topology is restricted
to the space of functions that are continuous when points are approached from
a certain direction (see, Bucher et al., (2014)). More precisely, two
functions are close in the $J_{1}$ topology only if, up to a small
perturbation in the coordinates, their supremum distance is small. This
implies that their jumps, if any, must match both in location and in
magnitude. Since, without loss of generality a sequence of continuous
functions cannot converge to a discontinuous function, we cannot use arguments
such as the hypiconvergence.

We establish weak convergence of the joint functional $L_{n}:=(L_{1n},L_{2n})$
based on weak convergence arguments of the corresponding point process of
jumps, for partial sum processes of stationary regularly varying sequences in
the $M_{1}$ topology (see, Ledger, (2016)). Using the connection between Lévy
processes and infinitely divisible distributions as discussed in the study of
Upadhye and Barman, (2022) and Krizmanić, (2018) (see, also Jacod and Protter,
(2011)), we demonstrate that these convergence results hold for cases in which
the tail index satisfies $\alpha\in(0,2)$. Our main result in relation to the
joint weak convergence is presented in our theorems which show that the
particular topological convergence holds for a more general dependence
structure than the standard i.i.d setting regardless of the number of
observations in each cluster.

The rest of the paper is organized as follows. Section 2 discusses the
relevant preliminary theory related to regular variation and the forms of weak
and strong $M_{1}$ topologies. Section 3 presents the main results of our
framework on weak convergence of weak convergence of self-normalized partial-
sum processes for the cases in which $\alpha\in(0,1)$ and $\alpha\in[1,2)$.
Section 4 illustrates theoretical applications of the main results to the
construction of self-normalized partial-sum processes for stationary regularly
varying sequences. All main proofs can be found in the Appendix of the paper.

##  Stationary regularly varying sequences

In this section we present related preliminary theory which apply specifically
to stationary regularly varying sequences, following Krizmanić, (2014, 2016,
2017) and Basrak and Krizmanić, (2015).

###  Regular variation

Let $\mathbb{E}^{d}=[-\infty,\infty]^{d}\setminus\\{0\\}$. We equip
$\mathbb{E}^{d}$ with the topology in which a set $B\subset\mathbb{E}^{d}$ has
compact closure if and only if it is bounded away from zero, if there exists
$u>0$ such that $B\subset\mathbb{E}^{d}_{u}=\\{x\in\mathbb{E}^{d}:\|x\|>u\\}$.
Here $\|\cdot\|$ denotes the max-norm on $\mathbb{R}^{d}$, i.e.
$\displaystyle\|x\|=\max\\{|x_{i}|:i=1,\ldots,d\\}$ where
$x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$. Denote by
$C_{K}^{+}(\mathbb{E}^{d})$ the class of all non-negative, continuous
functions on $\mathbb{E}^{d}$ with compact support.

######  Definition 1.

A stationary process $(X_{n})_{n\in\mathbb{Z}}$ is _(jointly) regularly
varying_ with index $\alpha\in(0,\infty)$ if for any nonnegative integer $k$
the $kd$-dimensional random vector $X=(X_{1},\ldots,X_{k})$ is multivariate
regularly varying with index $\alpha$, i.e. there exists a random vector
$\Theta$ on the unit sphere
$\mathbb{S}^{kd-1}=\\{x\in\mathbb{R}^{kd}:\|x\|=1\\}$ such that for every
$u\in(0,\infty)$ and as $x\to\infty$,

(2.1)
$\frac{\mathbb{P}(\|X\|>ux,\,X/\|X\|\in\cdot\,)}{\mathbb{P}(\|X\|>x)}\xrightarrow{w}u^{-\alpha}\mathbb{P}(\Theta\in\cdot\,),$

Then, regular variation can be expressed in terms of vague convergence of
measures on $\mathbb{E}$

(2.2) $n\mathbb{P}(a_{n}^{-1}X_{i}\in\cdot\,)\xrightarrow{v}\mu(\,\cdot\,),$

where $(a_{n})_{n\in\mathbb{N}}$ is a sequence of positive real numbers such
that the following condition holds

(2.3) $n\mathbb{P}(|X_{1}|>a_{n})\to 1$

as $n\to\infty$, and $\mu$ is a nonzero Radon measure on $\mathbb{E}$ given by

(2.4)
$\mu(\mathrm{d}x)=\bigl{(}p\,1_{(0,\infty)}(x)+q\,1_{(-\infty,0)}(x)\bigr{)}\,\alpha|x|^{-\alpha-1}\,\mathrm{d}x,$

for some $p\in[0,1]$, with $q=1-p$.

Specifically, Theorem 2.1 in Basrak and Segers, (2009) provides a convenient
characterization of joint regular variation: it is necessary and sufficient
that there exists a process $(Y_{n})_{n\in\mathbb{Z}}$ with
$\Pr(|Y_{0}|>y)=y^{-\alpha}$ for $y\geq 1$ such that, as $x\to\infty$,

(2.5) $\bigl{(}(x^{-1}\
X_{n})_{n\in\mathbb{Z}}\,\big{|}\,|X_{0}|>x\bigr{)}\xrightarrow{\text{fidi}}(Y_{n})_{n\in\mathbb{Z}},$

where "$\xrightarrow{\text{fidi}}$" denotes convergence of finite-dimensional
distributions. The process $(Y_{n})$ is called the _tail process_ of
$(X_{n})_{n\in\mathbb{N}}$.

###  Point processes and dependence conditions

Let $(X_{n})_{n\in\mathbb{N}}$ be a stationary sequence of random variables
and assume it is jointly regularly varying with index $\alpha>0$. Let
$(Y_{n})_{n\in\mathbb{Z}}$ be the tail process of $(X_{n})_{n\in\mathbb{Z}}$.
Suppose that the stochastic process $N_{n}$ is defined as below

$N_{n}=\sum_{i=1}^{n}\delta_{(i/n,\,X_{i}/a_{n})}\qquad\textrm{for all}\
n\in\mathbb{N},$

where the norming sequence $\left\\{a_{n}\right\\}$ is defined such that the
condition of expression (2.3) holds.

The point process convergence for the sequence $(N_{n})$ has been established
by Basrak et al., (2010) on the space $[0,1]\times\mathbb{E}_{u}$ for any
threshold $u>0$, with the limit depending on the threshold. A useful
convergence result for $N_{n}$ without the restrictions to various domains was
obtained by Basrak and Tafro, (2016). Appropriate weak dependence conditions
for this convergence result are given below. Both Condition 1 and Condition 2
provide necessary regularity assumptions.

######  Condition 1.

There exists a sequence of positive integers $(r_{n})$ such that
$r_{n}\to\infty$ and $r_{n}/n\to 0$ as $n\to\infty$ and such that for every
$f\in C_{K}^{+}([0,1]\times\mathbb{E})$, denoting $k_{n}=\lfloor
n/r_{n}\rfloor$, as $n\to\infty$,

(2.6)
$\mathbf{E}\biggl{[}\exp\biggl{\\{}-\sum_{i=1}^{n}f\biggl{(}\frac{i}{n},\frac{X_{i}}{a_{n}}\biggr{)}\biggr{\\}}\biggr{]}-\prod_{k=1}^{k_{n}}\mathbf{E}\biggl{[}\exp\biggl{\\{}-\sum_{i=1}^{r_{n}}f\biggl{(}\frac{kr_{n}}{n},\frac{X_{i}}{a_{n}}\biggr{)}\biggr{\\}}\biggr{]}\to
0.$

######  Condition 2.

There exists a sequence of positive integers $(r_{n})$ such that
$r_{n}\to\infty$ and $r_{n}/n\to 0$ as $n\to\infty$ and such that for every
$u>0$,

(2.7)
$\lim_{m\to\infty}\limsup_{n\to\infty}\mathbb{P}\biggl{(}\max_{m\leq|i|\leq
r_{n}}|X_{i}|>ua_{n}\,\bigg{|}\,|X_{0}|>ua_{n}\biggr{)}=0.$

######  Remark 1.

Condition 1 is implied by the strong mixing property (see Krizmanić, (2016)),
with the sequence $(\xi_{n})_{n\in\mathbb{N}}$ being a strongly mixing
sequence if $\alpha(n)\to 0$ as $n\to\infty$, where

$\displaystyle\alpha(n)=\sup\bigg{\\{}\big{|}\mathbb{P}(A\cap
B)-\mathbb{P}(A)\mathbb{P}(B)\big{|}:A\in\mathcal{F}_{-\infty}^{0},B\in\mathcal{F}_{n}^{\infty}\bigg{\\}}$

and $\mathcal{F}_{k}^{l}=\sigma(\\{\xi_{i}:k\leq i\leq l\\})$ for $-\infty\leq
k\leq l\leq\infty$ such that $\alpha(n)$ represents the mixing coefficient.

For a detailed discussion on joint regular variation and the dependence
properties of Condition 1 and Condition 2 we refer to Section 3.4 in Basrak et
al., (2010). These conditions are applicable to various time series regression
settings which include among others, moving averages processes333Further
applications presented in the literature which could be extended based on the
regularity conditions we provide here include ARMAX processes (see, Ferreira
and Ferreira, (2013)) as well as residual marked empirical processes of the
form
$\alpha_{n}(x)=\sum_{i=1}^{n}\mathsf{g}\left(X_{i-1}/a_{n}\right)\left(\mathbf{1}\left\\{\epsilon_{i}\leq
x\right\\}-F(x)\right)$ (see, Chan and Zhang, (2013)), which are used for
constructing goodness-of-fit tests., stochastic volatility models as well as
GARCH-type processes (see, Section 4 in Basrak et al., (2010)). Notice that
Condition 1 presented above is as Condition 2.2. in Krizmanić, (2014),
Condition 2.1 in Krizmanić, (2016) and Condition 2.1 in Krizmanić, (2017).

In order to present the main weak convergence results for the pair
$(L_{1n},L_{2n})$ we need to define some relevant terms used in our analysis
of random regularly varying sequences. By Proposition 4.2 in Basrak and Tafro,
(2016), under Condition 2 the following result holds

(2.8) $\displaystyle\theta:=\mathbb{P}\left(\underset{i\geq
1}{\mathsf{sup}}|Y_{i}|\leq
1\right)=\mathbb{P}\left(\underset{i\leq-1}{\mathsf{sup}}|Y_{i}|\leq
1\right)>0,$

where $\theta$ denotes the extremal index of the univariate sequence
$(|X_{n}|)_{n\in\mathbb{N}}$.

Under the assumption of joint regular variation and Condition 1 and 2, by
Theorem 3.1 in Basrak and Tafro Basrak and Tafro, (2016), the following
convergence in distribution holds,

(2.9)
$N_{n}\xrightarrow{d}N:=\sum_{i}\sum_{j}\delta_{(T_{i},P_{i}\eta_{ij})}\in[0,1]\times\mathbb{E}\
\ \ \text{as}\ n\to\infty,$

where $\sum_{i=1}^{\infty}\delta_{(T_{i},P_{i})}$ is a Poisson process on
$[0,1]\times(0,\infty)$ with intensity measure $Leb\times\nu$.

Moreover, $\nu(\mathrm{d}x)=\theta\alpha
x^{-\alpha-1}\boldsymbol{1}_{(0,\infty)}(x)\,\mathrm{d}x$ and
$(\sum_{j=1}^{\infty}\delta_{\eta_{ij}})_{i}$ is an i.i.d. sequence of point
processes in $\mathbb{E}$ independent of $\sum_{i}\delta_{(T_{i},P_{i})}$ with
common distribution equal to the distribution of
$\sum_{j}\delta_{Z_{j}/L_{Z}}$, where $L_{Z}=\sup_{j\in\mathbb{Z}}|Z_{j}|$ and
$\sum_{j}\delta_{Z_{j}}$ is distributed as
$(\sum_{j\in\mathbb{Z}}\delta_{Y_{j}}\,|\,\sup_{i\leq-1}|Y_{i}|\leq 1)$.

######  Corollary 1.

(Davis and Hsing, (1995)) Let $\sum_{j}\delta_{\eta_{j}}$ be a point process
with a limit distribution equal to the distribution of
$\sum_{j}\delta_{\eta_{1j}}$. Then, for $\alpha\leq 1$ it holds that

(2.10) $\mathbf{E}\Big{(}\sum_{j}|\eta_{j}|\Big{)}^{\alpha}<\infty.$

######  Remark 2.

The finite moment condition given by Corollary 1 provides a necessary
condition for the weak convergence of point processes into a stable
distribution when $\alpha\leq 1$. Although the particular finiteness of
moments condition of point processes might not hold for $\alpha>1$ (see
Mikosch and Wintenberger, (2014)).

###  The weak and strong $M_{1}$ topologies

In this section, we introduce the definition of the Skorokhod weak $M_{1}$
topology in a general space $D([0,1],\mathbb{R}^{d})$ of
$\mathbb{R}^{d}$–valued càdlàg functions on the closed set $[0,1]$. The
definitions presented below can be found in Section 3 of Basrak and Krizmanić,
(2015) and in Section 2.3 of Krizmanić, (2017).

######  Definition 2 (Completed graph).

For $x\in D([0,1],\mathbb{R}^{d})$ the completed (thick) graph of $x$ is the
set

(2.11) $\displaystyle
G_{x}=\big{\\{}(t,z)\in[0,1]\times\mathbb{R}^{d}:z\in[[x(t-),x(t)]]\big{\\}},$

where $x(t-)$ is the left limit of $x$ at $t$ and $[[a,b]]$ is the product
segment, such that

(2.12)
$\displaystyle[[a,b]]=[a_{1},b_{1}]\times[a_{2},b_{2}]\ldots\times[a_{d},b_{d}]$

where $a=(a_{1},a_{2},\ldots,a_{d})$ and
$b=(b_{1},b_{2},\ldots,b_{d})\in\mathbb{R}^{d}$.

######  Definition 3 (Order of graph).

An order on the graph $G_{x}$ is defined by assuming that
$(t_{1},z_{1})\leq(t_{2},z_{2})$ if either (i) $t_{1}<t_{2}$ or (ii)
$t_{1}=t_{2}$ and $|x_{j}(t_{1}-)-z_{1j}|\leq|x_{j}(t_{2}-)-z_{2j}|$, which
holds for all $j=1,2,\ldots,d$.

######  Definition 4 (Partial Order of graph).

The relation $\preceq$ induces only a partial order on the graph $G_{x}$. A
weak parametric representation of the graph $G_{x}$ is a continuous non-
decreasing function $(r,u)$ mapping $[0,1]$ into $G_{x}$, with $r\in
C([0,1],[0,1])$ being the time component and $u\in C([0,1],\mathbb{R}^{d})$
being the spatial component, such that $r(0)=0,r(1)=1$ and $u(1)=x(1)$.

######  Definition 5.

Let $\Pi_{w}(x)$ denote the set of weak parametric representations of the
graph $G_{x}$. For $x_{1},x_{2}\in D([0,1],\mathbb{R}^{d})$ define

(2.13) $\displaystyle
d_{w}(x_{1},x_{2})=\inf\big{\\{}\|r_{1}-r_{2}\|_{[0,1]}\vee\|u_{1}-u_{2}\|_{[0,1]}:(r_{i},u_{i})\in\Pi_{w}(x_{i}),i=1,2\big{\\}},$

where $\|x\|_{[0,1]}=\sup\\{\|x(t)\|:t\in[0,1]\\}$.

######  Definition 6.

(Weak $M_{1}$ topology) We say that $x_{n}\to x$ in $D([0,1],\mathbb{R}^{d})$
for a sequence $(x_{n})_{n\in\mathbb{R}}$ in the weak Skorokhod $M_{1}$
topology if $d_{w}(x_{n},x)\to 0$ as $n\to\infty$.

######  Definition 7.

(Strong $M_{1}$ topology) For $x\in D([0,1],\mathbb{R}^{d})$ we let

(2.14)
$\displaystyle\Gamma_{x}=\big{\\{}(t,z)\in[0,1]\times\mathbb{R}^{d}:z\in[x(t-),x(t)]\big{\\}},$

where $[a,b]=\\{\lambda a+(1-\lambda)b:0\leq\lambda\leq 1\\}$ for
$a,b\in\mathbb{R}^{d}$. We say that $(r,u)$ is a strong parametric
representation of $\Gamma_{x}$ if it is a continuous non-decreasing function
mapping $[0,1]$ onto $\Gamma_{x}$.

Denote by $\Pi(x)$ the set of all strong parametric representations of the
completed (thin) graph $\Gamma_{x}$. Then for $x_{1},x_{2}\in
D([0,1],\mathbb{R}^{d})$ denote with

(2.15) $\displaystyle
d_{M_{1}}(x_{1},x_{2})=\inf\big{\\{}\|r_{1}-r_{2}\|_{[0,1]}\vee\|u_{1}-u_{2}\|_{[0,1]}:(r_{i},u_{i})\in\Pi(x_{i}),i=1,2\big{\\}}.$

Then, we say that the distance function $d_{M_{1}}(x_{1},x_{2})$ is a metric
on $D([0,1],\mathbb{R}^{d})$, and the induced topology is called the (standard
or strong) Skorokhod $M_{1}$ topology.

######  Remark 3.

Notice that the $WM_{1}$ topology is weaker than the standard $M_{1}$ topology
on $D([0,1],\mathbb{R}^{d})$, but they coincide in the case that $d=1$.
Furthermore, the $WM_{1}$ topology coincides with the topology induced by the
metric

(2.16)
$d_{p}(x_{1},x_{2})=\max\big{\\{}d_{M_{1}}(x_{1j},x_{2j}):j=1,2,\ldots,d\big{\\}}$

for $x_{i}=(x_{i1},x_{i2},\ldots,x_{id})\in D([0,1],\mathbb{R}^{d})$ and
$i=1,2$. Then, we say that the metric $d_{p}(x_{1},x_{2})$ induces the product
topology on $D([0,1],\mathbb{R}^{d})$. For detailed discussion of the strong
and weak $M_{1}$ topologies we refer to sections 12.3–12.5 in Whitt, (2002).

Next, we use the above definitions in order to introduce the induced
measurability results. Denote by $C^{\uparrow}_{0}([0,1],\mathbb{R})$ the
subset of functions $x$ in the space $D([0,1],\mathbb{R})$ that are continuous
and nondecreasing which satisfy $x(0)>0$. Therefore, we assume that
$C^{\uparrow}_{0}([0,1],\mathbb{R})$ is a measurable subset of
$D([0,1],\mathbb{R})$ with the Borel $\sigma$–fields associated with the
Skorokhod $J_{1}$ and $M_{1}$ topologies, which coincide with the Kolmogorov
$\sigma$–field generated by the projection maps (see Theorem 11.5.2 and
Theorem 11.5.3 in Whitt, (2002)).

######  Remark 4.

We only consider weak convergence of partial-sum processes in the subset
$C^{\uparrow}_{n}([0,1],\mathbb{R})$. Notice that the $J_{1}$ topology
corresponds to a closed set while the standard $M_{1}$ topology corresponds to
an open set, which it is still assumed to be a measurable set. As a result,
the set $C^{\uparrow}_{n}([0,1],\mathbb{R})$ corresponds to the subset of
càdlàg functions which corresponds to continuous, positive and non-decreasing
functions, which ensures the measurability conditions.

######  Lemma 1.

The set $C^{\uparrow}_{0}([0,1],\mathbb{R})$ is a measurable subset of
$D[0,1],\mathbb{R})$ with the $M_{1}$ topology.

###### Proof.

The proof of 1 can be found in the Appendix. ∎

Consequently, to facilitate the development of limit theory for the proposed
framework, we introduce the concept of $M_{1}$ continuity of multiplication
and division of two càdlàg functions which are well-established results in the
literature. Specifically, the first result presented in Lemma 2 is based on
Theorem 13.3.2 of Whitt, (2002), while the second result presented in Lemma 3
is based on the fact that for monotone functions $M_{1}$ convergence is
equivalent to point-wise convergence in a dense subset of $[0,1]$ including
$0$ and $1$ (see Whitt, (2002), Corollary 12.5.1).

Denote by $\textrm{Disc}(x)$ the set of discontinuity points of $x\in
D([0,1],\mathbb{R})$.

######  Lemma 2.

Suppose that $x_{n}\to x$ and $y_{n}\to y$ in $D([0,1],\mathbb{R})$ with the
$M_{1}$ topology. If it holds that for each
$t\in\textrm{Disc}(x)\cap\textrm{Disc}(y)$, $x(t)$, $x(t-)$, $y(t)$ and
$y(t-)$ are all nonnegative and $[x(t)-x(t-)][y(t)-y(t-)]\geq 0$, then

(2.17) $\displaystyle x_{n}y_{n}\to xy,\ \ \ \in D([0,1],\mathbb{R})$

with the $M_{1}$ topology, where $(xy)(t)=x(t)y(t)$ for $t\in[0,1]$.

######  Lemma 3.

Consider the function $h\colon D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})\to D([0,1],\mathbb{R})$ defined by
$h(x,y)=\frac{x}{y}$, where

(2.18) $\displaystyle\Big{(}\frac{x}{y}\Big{)}(t)=\frac{x(t)}{y(t)},\qquad
t\in[0,1].$

Then the function h is continuous when $D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})$ is endowed with the weak $M_{1}$ topology
and $D([0,1],\mathbb{R})$ is endowed with the standard $M_{1}$ topology.

###### Proof.

The proof of Lemma 3 can be found in the Appendix. ∎

The next section considers conditions under which the joint partial sum
process $L_{n}$ satisfies a functional limit theorem with the limit
$L=\left(L_{1},L_{2}\right)$, where $L_{1}$ is an $\alpha-$stable Lévy process
and $L_{2}$ is is an $\alpha/2-$stable Lévy process. Specifically, we express
the components of the limiting process $L=\left(L_{1},L_{2}\right)$ as
functionals of the limiting point process
$N=\sum_{i}\sum_{j}\delta_{(T_{i},P_{i}\eta_{ij})}$ based on the
characteristic triple of $L_{1}$, given by $(a,\nu,b)$. Notice that
$a\in\mathbb{R}$ is the location parameter and $\alpha\in(0,2)$ is the
stability parameter, which is useful in determining the decay of the tail of
the distribution of $X$. Specifically, we consider cases such that $\alpha<2$,
which excludes Gaussian processes. We introduce a mode of convergence for
real-valued, locally bounded functions on a locally compact metrizable space
which does not necessarily require that the i.i.d assumption holds, thereby
allowing for clustering of dependent observations for stationary regularly
varying sequences. Our main result establishes the weak convergence of self-
normalized partial sum processes of stationary regularly varying sequences to
a stable Lévy process in the space $D[0,1]$ equipped with Skorokhod’s $M_{1}$
topology.

##  Joint Functional Convergence of Partial Sum Processes

###  Joint Functional Convergence of Partial Sum Processes with
$\alpha\in(0,1)$

Let $(X_{n})_{n\in\mathbb{Z}}$ be a stationary sequence of random variables,
jointly regularly varying with index $\alpha\in(0,1)$. Assume Conditions 1 and
2 hold. Let

(3.1) $\displaystyle S_{n}=\sum_{k=1}^{n}X_{k},\qquad
V^{2}_{n}=\sum_{k=1}^{n}X_{k}^{2},\qquad n\in\mathbb{N}$

and denote with $L_{n}(\cdot)$, the joint partial sum process where
$L_{n}(\cdot):=\big{(}L_{1n}(\cdot),L_{2n}(\cdot)\big{)}$ such that

(3.2) $\displaystyle
L_{n}(t)=\left(\sum_{k=1}^{[nt]}\frac{X_{k}}{a_{n}},\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X_{k}^{2}}{a_{n}^{2}}\right),\qquad t\in[0,1],$

where the sequence $(a_{n})_{n\in\mathbb{N}}$ defined by expression (2.2) such
that property (2.3) holds.

The first coordinate of the joint functional $L_{n}(.)$ corresponds to partial
sums of a linear process while the second coordinate corresponds to partial
sums of squares of a linear process. The key idea to establish weak
convergence results on a probability space
$\left(\Omega,\mathbb{P},\mathcal{F}\right)$, as $n\to\infty$, is to represent
the joint stochastic process $L_{n}$ as the image of the time-space point
process $N_{n}$ under an appropriate summation functional. Using the
continuity properties of $L_{n}$ and by an application of the continuous
mapping theorem we can transfer the weak convergence of the time-space point
process $N_{n}$, given by (2.9) to weak convergence of the joint stochastic
process $L_{n}$ in the $M_{1}$ topology. Fix $0<u<\infty$ and define the
summation functional

(3.3) $\displaystyle\Phi^{(u)}\colon\mathbf{M}_{p}([0,1]\times\mathbb{E})\to
D([0,1],\mathbb{R}^{2})$ (3.4)
$\displaystyle\Phi^{(u)}\Big{(}\sum_{i}\delta_{(t_{i},x_{i})}\Big{)}(t)=\Big{(}\sum_{t_{i}\leq
t}x_{i}\,1_{\\{u<|x_{i}|<\infty\\}},\sum_{t_{i}\leq
t}x_{i}^{2}\,1_{\\{u<|x_{i}|<\infty\\}}\Big{)},\qquad t\in[0,1],$

where the space $\mathbf{M}_{p}([0,1]\times\mathbb{E})$ of Radon point
measures on $[0,1]\times\mathbb{E}$ is equipped with the vague topology (see
Resnick, (2008), Chapter 3). Let $\Lambda=\Lambda_{1}\cap\Lambda_{2}$, where

(3.5)
$\Lambda_{1}=\\{\eta\in\mathbf{M}_{p}([0,1]\times\mathbb{E}):\eta(\\{0,1\\}\times\mathbb{E})=0=\eta([0,1]\times\\{\pm\infty,\pm
u\\})\\},\\\\[10.00002pt]
\Lambda_{2}=\\{\eta\in\mathbf{M}_{p}([0,1]\times\mathbb{E}):\eta(\\{t\\}\times(u,\infty])\cdot\eta(\\{t\\}\times[-\infty,-u))=0,\
\text{for all $t\in[0,1]$}\\}.$

The definitions of the sets $\Lambda_{1}$ and $\Lambda_{2}$ are instrumental
for our main results. The elements of $\Lambda_{2}$ have the property that
atoms in $[0,1]\times\mathbb{E}_{u}$ with the same time coordinate are all on
the same side of the time axis. Similar to Lemma 3.1 in Basrak et al., (2010)
one can prove that the Poisson cluster process $N$ given by (2.9) a.s. belongs
to the set $\Lambda$ given that each cluster of large values of
$(X_{n})_{n\in\mathbb{N}}$ contains only values of the same sign. The link
between the tail measure of a regularly varying time series and its spectral
tail process is studied by Dombry et al., (2018).

######  Lemma 4.

Assume that with probability one the tail process $(Y_{i})_{i\in\mathbb{Z}}$
of the sequence $(X_{n})_{n\in\mathbb{N}}$ has no two values of the opposite
sign. Then it holds that,

(3.6) $\displaystyle\mathbb{P}(N\in\Lambda)=1.$

Next, we examine the continuity properties of the stochastic process
$\Phi^{(u)}$ is over the set $\Lambda$, where
$\Lambda=\Lambda_{1}\cap\Lambda_{2}$ and $\Lambda_{1},\Lambda_{2}$ are given
by (Step 1.).

######  Lemma 5.

The summation functional
$\Phi^{(u)}\colon\mathbf{M}_{p}([0,1]\times\mathbb{E})\to
D([0,1],\mathbb{R}^{2})$ is continuous on the set $\Lambda$, when
$D([0,1],\mathbb{R}^{2})$ is endowed with the weak $M_{1}$ topology.

In the theorem below we establish functional convergence of the stochastic
process $L_{n}$, with the limit $L=(L_{1},L_{2})$ consisting of stable Lévy
processes $L_{1}$ and $L_{2}$. Without loss of generality, the distribution of
a Lévy process $V$ is characterized by its characteristic triple, that is, the
characteristic triple of the infinitely divisible distribution of $V(1)$.
Then, the characteristic function of $V(1)$ and the characteristic triple
$(a,\nu,b)$ are related in the following way:

(3.7)
$\mathbf{E}\left[e^{izV(1)}\right]=\exp\biggl{(}-\frac{1}{2}az^{2}+ibz+\int_{\mathbb{R}}\bigl{(}e^{izx}-1-izx\boldsymbol{1}_{[-1,1]}(x)\bigr{)}\,\nu(\mathrm{d}x)\biggr{)}$

for some $z\in\mathbb{R}$. Here $a\geq 0$, $b\in\mathbb{R}$ are constants, and
$\nu$ is a measure on $\mathbb{R}$ satisfying

(3.8)
$\displaystyle\nu(\\{0\\})=0\qquad\text{and}\qquad\int_{\mathbb{R}}(|x|^{2}\wedge
1)\,\nu(\mathrm{d}x)<\infty.$

######  Theorem 1 (see, Krizmanić, (2020)).

Let $(X_{n})_{n\in\mathbb{N}}$ be a stationary sequence of random variables,
jointly regularly varying with index $\alpha\in(0,1)$, and of which the tail
process $(Y_{i})_{i\in\mathbb{Z}}$ almost surely has no two values of the
opposite sign. Suppose that Conditions 1 and 2 hold. Then it holds that,

(3.9) $\displaystyle L_{n}\xrightarrow{d}L\qquad\textrm{as}\ n\to\infty,$

in $D([0,1],\mathbb{R}^{2})$ endowed with the weak $M_{1}$ topology, where
$L=(L_{1},L_{2})$, $L_{1}$ is an $\alpha$–stable Lévy process with
characteristic triple $(0,\nu_{1},\gamma_{1})$ and $L_{2}$ is an
$\alpha/2$–stable Lévy process with characteristic triple
$(0,\nu_{2},\gamma_{2})$, where

(3.10) $\displaystyle\nu_{1}(\mathrm{d}x)$
$\displaystyle=\theta\alpha\big{(}c_{+}1_{(0,\infty)}(x)+c_{-}1_{(-\infty,0)}(x)\big{)}|x|^{-\alpha-1}\,\mathrm{d}x,$
(3.11) $\displaystyle\nu_{2}(\mathrm{d}x)$
$\displaystyle=\frac{\theta\alpha}{2}\mathbf{E}\bigg{(}\sum_{j}\eta_{j}^{2}\bigg{)}^{\alpha/2}x^{-\alpha/2-1}1_{(0,\infty)}(x)\,\mathrm{d}x,$
(3.12) $\displaystyle c_{+}$
$\displaystyle=\mathbf{E}\bigg{[}\Big{(}\sum_{j}\eta_{j}\Big{)}^{\alpha}1_{\\{\sum_{j}\eta_{j}>0\\}}\bigg{]},\quad
c_{-}=\mathbf{E}\bigg{[}\Big{(}-\sum_{j}\eta_{j}\Big{)}^{\alpha}1_{\\{\sum_{j}\eta_{j}<0\\}}\bigg{]}$
(3.13) $\displaystyle\gamma_{1}$
$\displaystyle=\frac{\theta\alpha}{1-\alpha}(c_{+}-c_{-}),\qquad\gamma_{2}=\frac{\theta\alpha}{2-\alpha}\mathbf{E}\bigg{(}\sum_{j}\eta_{j}^{2}\bigg{)}^{\alpha/2}.$

######  Remark 5.

Recall that a Levy process $V$ with the characteristic triple given by
$(b,c,\nu)$, with respect to a truncation function $\kappa$, is a stochastic
process with a characteristic function given by

(3.14)
$\displaystyle\mathbb{E}\big{[}e^{izL_{t}}\big{]}=\mathsf{exp}\left(-\frac{1}{2}tcz^{2}-itbz+t\int_{\mathbb{R}}\big{(}e^{izx}-1-iz\kappa(x)\big{)}\nu(dx)\right),$

where $\kappa(x)=-\kappa(x)$ and $t\geq 0$, with
$\kappa(x)=\boldsymbol{1}_{[-1,1]}$. For example, if $t=1$, then we have that

(3.15)
$\displaystyle\mathbb{E}\big{[}e^{i\mathsf{z}L(1)}\big{]}=\mathsf{exp}\left(-\frac{1}{2}cz^{2}-ibz+\int_{\mathbb{R}}\big{(}e^{izx}-1-iz\kappa(x)\big{)}\nu(dx)\right),$

For infinite order linear processes with all coefficients of the same sign the
idea is to approximate them by a sequence of finite order linear processes for
which the theoretical conditions of the main theorem applies to the case of
finite order linear processes with a truncation parameter $m$. Then, in order
to verify that the equivalence limit process result holds, we need to
determine the exact values of the characteristic triple process $V$. Moreover,
to complete the proof we need to check that the stochastic process
$L_{n}(t):=\big{(}V_{n}(t),W_{n}(t)\big{)}$ jointly weakly converges within
the $M_{1}$ topology which implies that each coordinate separately satisfies
these theoretical assumptions and weakly convergence arguments. Then, proving
that $L_{n}$ converge in distribution to some asymptotic process, say
$L=\big{(}V,W\big{)}$, in the standard $M_{1}$ topology, then using the fact
that linear combinations of the coordinates of the joint functional are
continuous and satisfy all related assumptions in the same topological space,
then we can conclude that indeed $L_{n}$ weakly converge to its asymptotic
counterpart $L$, endowed with the $M_{1}$ topology.

###  Joint Functional Convergence of Partial Sum Processes with
$\alpha\in[1,2)$

In this section we concentrate on establishing the joint functional weak
convergence result for the case where the regularly varying index
$\alpha\in[1,2)$. In other words, we investigate the joint weak functional
limiting behaviour of the partial sum and partial sum squares processes given
by the stochastic process $L_{n}(.)=\left(L_{1n}(.),L_{2n}(.)\right)$ in the
space $D\left([0,1],\mathbb{R}^{2}\right)$, where

(3.16) $\displaystyle L_{1n}(t)=\frac{S_{\lfloor nt\rfloor}}{a_{n}}-\lfloor
nt\rfloor b_{1n},\ \ \ L_{2n}(t)=\frac{V^{2}_{\lfloor
nt\rfloor}}{a_{n}}-\lfloor nt\rfloor b_{2n},\ \ \ t\in[0,1],$

where $S_{\lfloor nt\rfloor}=\sum_{k=1}^{\lfloor nt\rfloor}X_{k}$ and
$V^{2}_{\lfloor nt\rfloor}=\sum_{k=1}^{\lfloor nt\rfloor}X^{2}_{k}$. Since
$\left\\{a_{n}\right\\}$ is a sequence of positive real numbers such that
$n\mathbb{P}\left(\left|X_{1}\right|>a_{n}\right)\to 1$, as $n\to\infty$ and
$L_{n}$ correspond to the centered stochastic process
$L_{n}=\big{(}L_{1n},L_{2n}\big{)}$ by the expected value of the truncated
functional of each coordinate.

Therefore, we obtain the following expressions

$\displaystyle L_{1n}(t)$ $\displaystyle=\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X_{k}}{a_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{X_{1}}{a_{n}}\mathbf{1}_{\left\\{u<\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right],\ \ \text{for}\ \alpha\in[1,2)$ $\displaystyle L_{2n}(t)$
$\displaystyle=\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X^{2}_{k}}{a^{2}_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\left\\{u<\frac{X_{1}^{2}}{a^{2}_{n}}\leq
1\right\\}}\right],\ \ \text{for}\ \alpha\in[1,2)$

The constants $b_{1n}$ and $b_{2n}$ are defined with respect to the the
regularly varying index as below

(3.17) $\displaystyle b_{1n}$
$\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\mathbb{E}\left[\frac{X_{1}}{a_{n}}\mathbf{1}_{\left\\{\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right],&\alpha\in[1,2).\end{cases}$ (3.18) $\displaystyle b_{2n}$
$\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\mathbb{E}\left[\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\left\\{\frac{X^{2}_{1}}{a^{2}_{n}}\leq
1\right\\}}\right],&\alpha\in[1,2).\end{cases}$

######  Remark 6.

Based on the functional convergence result given above we can show that the
weak limit of self-normalized partial-sum processes is indeed a stable Lévy
process where $\left\\{b_{n}\right\\}$ is a sequence (see, (3.17) and (3.18))
such that it induces centered random variables for the stochastic processes
under study. Specifically, for the self-normalized partial sum process, we
need to prove that $S_{\lfloor
n.\rfloor}/V_{n}\overset{d}{\to}L_{1}(\,\cdot\,)\big{/}\sqrt{L_{2}(1)}$, is an
$\alpha-$stable Lévy process

(3.19) $\displaystyle\frac{S_{\lfloor
n.\rfloor}}{V_{n}}\overset{d}{\to}\frac{L_{1}(\,\cdot\,)}{\sqrt{L_{2}(1)}}:=\mathcal{L}_{\alpha}(\,\cdot\,)$

where the limit $\mathcal{L}_{\alpha}(\,\cdot\,)$ represents an
$\alpha-$stable Lévy process (see, Theorem 2 and Theorem 4).

By imposing the vanishing small values condition then convergence of the
normalized partial sums can be established by showing that the limiting
distribution is infinitely divisible with a Lévy triplet corresponding to an
$\alpha-$stable distribution ( Denker and Jakubowski, (1989), Bartkiewicz et
al., (2011)). The functional convergence of $L_{n}$ in the special case when
$(X_{n})_{n\in\mathbb{N}}$ is a linear process from a regularly varying
distribution with index $\alpha\in(0,2)$ was studied in the literature
previously. Therefore, the main result of our article shows that for a
strictly stationary, regularly varying sequence of dependent random variables
$(X_{n})_{n\in\mathbb{N}}$ with index $\alpha\in(0,2)$, for which clusters of
high-threshold excesses can be broken down into asymptotically independent
blocks, the stochastic process $L_{n}$ converge in the space
$D\left([0,1],\mathbb{R}^{2}\right)$ endowed with the Skorokhod weak $M_{1}$
topology under the condition that all extremes within each cluster of big
values have the same sign. This topology is weaker than the more commonly used
Skorokhod $J_{1}$ topology, the latter being appropriate when there is no
clustering of extremes (i.i.d case).

Following similar argumentation as in the previous section, we will transfer
the weak convergence of $N_{n}$ in expression (2.9) to weak convergence of
$L_{n}$. When $\alpha\in[1,2)$ we need the following conditions to deal with
small jumps. Let $k=\lfloor\frac{n}{r}\rfloor$.

######  Condition 3.

For all $\delta>0$ it holds that,

(3.20) $\lim_{u\to 0}\limsup_{n\to\infty}\mathbb{P}\left(\max_{0\leq k\leq
n}\left|\sum_{i=1}^{k}\left(\frac{X_{i}}{a_{n}}\mathbf{1}_{\left\\{\frac{|X_{i}|}{a_{n}}\leq
u\right\\}}-\mathbf{E}\left[\frac{X_{i}}{a_{n}}\mathbf{1}_{\left\\{\frac{|X_{i}|}{a_{n}}\leq
u\right\\}}\right]\right)\right|>\delta\right)=0.$

######  Condition 4.

For all $\delta>0$ it holds that,

(3.21) $\lim_{u\to 0}\limsup_{n\to\infty}\mathbb{P}\left(\max_{0\leq k\leq
n}\left|\sum_{i=1}^{k}\left(\frac{X_{i}^{2}}{a_{n}^{2}}\mathbf{1}_{\left\\{\frac{X_{i}^{2}}{a^{2}_{n}}\leq
u\right\\}}-\mathbf{E}\left[\frac{X_{i}^{2}}{a_{n}^{2}}\mathbf{1}_{\left\\{\frac{X_{i}^{2}}{a^{2}_{n}}\leq
u\right\\}}\right]\right)\right|>\delta\right)=0.$

######  Remark 7.

An equivalent of Condition 1 but for the sequence of squared random variables
holds, which we omit to present for space brevity. Moreover, Condition 1
provides a statement for dependent sequences for certain type of mixing
processes and it corresponds to the vanishing small values property. The index
$\alpha$ is considered as a measure of the regular variation property rather
than the degree of dependence of observations generated from the Lévy process,
thus the above conditions could be challenging to prove for different mixing
regimes. Condition 3 and 4 correspond to small jumps condition both for the
$\left(X_{n}\right)_{n\in\mathbb{R}}$ as well as
$\left(X^{2}_{n}\right)_{n\in\mathbb{R}}$ stationary, jointly regular varying
sequences. A similar condition is given by Assumption 4.4 in Basrak et al.,
(2018), which allows to prove that the finite dimensional marginal
distributions of $V_{n}$ converge to those of an $\alpha-$stable Lévy process.

######  Theorem 2.

Let $(X_{n})_{n\in\mathbb{N}}$ be a stationary sequence of random variables,
jointly regularly varying with index $\alpha\in[1,2)$, and of which the tail
process $(Y_{i})_{i\in\mathbb{Z}}$ almost surely has no two values of the
opposite sign. Suppose that Conditions 1 and 3 hold. Then, the stochastic
process

(3.22) $\displaystyle
L_{n}(t):=\big{(}L_{1n}(t),L_{2n}(t)\big{)}\equiv\left(\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X_{k}}{a_{n}}-\lfloor nt\rfloor b_{1n},\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X^{2}_{k}}{a^{2}_{n}}-\lfloor nt\rfloor b_{2n}\right),\ \ \
t\in[0,1],$

with $b_{1n}$ and $b_{2n}$ given by expressions (3.17) and (3.18)
respectively. Then, it holds that

(3.23) $\displaystyle L_{n}\xrightarrow{d}L\ \text{as}\ n\to\infty$

in $D([0,1],\mathbb{R}^{2})$ endowed with the weak $M_{1}$ topology, where
$L=\big{(}L_{1},L_{2}\big{)}$, $L_{1}$ is an $\alpha$–stable Lévy process with
characteristic triple $(0,\nu_{1},\gamma_{1})$ and $L_{2}$ is an
$\alpha/2$–stable Lévy process with characteristic triple
$(0,\nu_{2},\gamma_{2})$, where

(3.24) $\displaystyle\nu_{1}(\mathrm{d}x)$
$\displaystyle=\theta\alpha\big{(}c_{+}1_{(0,\infty)}(x)+c_{-}\boldsymbol{1}_{(-\infty,0)}(x)\big{)}|x|^{-\alpha-1}\,\mathrm{d}x,$
(3.25) $\displaystyle\nu_{2}(\mathrm{d}x)$
$\displaystyle=\frac{\theta\alpha}{2}\mathbf{E}\bigg{(}\sum_{j}\eta_{j}^{2}\bigg{)}^{\alpha/2}x^{-\alpha/2-1}\boldsymbol{1}_{(0,\infty)}(x)\,\mathrm{d}x,$
(3.26) $\displaystyle c_{+}$
$\displaystyle=\mathbf{E}\bigg{[}\Big{(}\sum_{j}\eta_{j}\Big{)}^{\alpha}\boldsymbol{1}_{\\{\sum_{j}\eta_{j}>0\\}}\bigg{]},\quad
c_{-}=\mathbf{E}\bigg{[}\Big{(}-\sum_{j}\eta_{j}\Big{)}^{\alpha}\boldsymbol{1}_{\\{\sum_{j}\eta_{j}<0\\}}\bigg{]}$
(3.27) $\displaystyle\gamma_{1}$
$\displaystyle=\frac{\alpha}{\alpha-1}\bigg{(}p-q-\theta\big{(}c_{+}-c_{-}\big{)}\bigg{)}.$

######  Remark 8.

Note that it can be proved that the two components of the limiting process
$L=(L_{1},L_{2})$ can be expressed as functionals of the limiting point
process $N=\sum_{i}\sum_{j}\delta_{(T_{i},P_{i}\eta_{ij})}$. In particular, we
define the limit process $L_{1}(\,\cdot\,)$ of the functional
$L_{1n}(\,\cdot\,)$ according to the value of the regular varying index
$\alpha$ such that

(3.28) $\displaystyle
L_{1}(\,\cdot\,)=\begin{cases}\displaystyle\sum_{T_{i}\leq.}\sum_{j}P_{i}\eta_{ij},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\sum_{T_{i}\leq.}\sum_{j}P_{i}\eta_{ij}1_{\left\\{P_{i}\left|\eta_{ij}\right|>u\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

where $L_{1}(\,\cdot\,)$ is an $\alpha-$stable Lévy process with
characteristic triple $(0,\nu_{1},\gamma_{1})$, where the limit in the latter
case holds almost surely uniformly on $[0,1]$ (along some sub-sequence).

Furthermore, we define the limit process $L_{2}(\,\cdot\,)$ of the functional
$L_{2n}(\,\cdot\,)$ according to the value of the regular varying index
$\alpha$ such that

(3.29) $\displaystyle
L_{2}(\,\cdot\,)=\begin{cases}\displaystyle\sum_{T_{i}\leq.}\sum_{j}P^{2}_{i}\eta^{2}_{ij},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\sum_{T_{i}\leq.}\sum_{j}P^{2}_{i}\eta^{2}_{ij}1_{\left\\{P^{2}_{i}\left|\eta_{ij}\right|>u\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x^{2}\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

where $L_{2}(\,\cdot\,)$ is an $\alpha/2-$stable Lévy process with
characteristic triple $(0,\nu_{2},\gamma_{2})$, where the limit in the latter
case holds almost surely uniformly on $[0,1]$ (along some sub-sequence).

###### Proof.

For the proof of Theorem 2 see Appendix B of the paper. ∎

######  Remark 9.

Notice that in the case when $\alpha\in[1,2)$, we assume that the contribution
of the smaller increments of the partial sum process is closer to its
expectation. The random variable $X_{1}^{2}$ has an infinite expectation, but
when we truncate this random variable with an appropriate indicator then we
obtain finite expectation such that the following expression is finite

(3.30)
$\displaystyle\mathbb{E}\left(\frac{X_{1}^{2}}{\alpha_{n}^{2}}\boldsymbol{1}_{\frac{|X_{1}|}{\alpha_{n}}\leq
1}\right)\leq 1<\infty.$

Moreover, for $\alpha\in[1,2)$ the functionals are suitable centered with the
coefficients $b_{1n}$ and $b_{2n}$ respectively. Then, the corresponding
convergence results for $\alpha\in[1,2)$ only hold for finite quantities.
Intuitively, Theorem 2 shows that the joint functional consisting of the
partial sum process in the first coordinate and the partial sum squared
process in the second coordinate satisfy a functional central limit theorem in
the weak $M_{1}$ topology with the limit consisting of an $\alpha-$stable Lévy
process in the first coordinate and an $\alpha/2-$stable Lévy process in the
second coordinate. In order to prove this result one needs to obtain the
analytical form of their characteristic triples.

###  Weak Convergence of Self-Normalized Partial Sum Processes

In this Section we establish the weak convergence of Self-normalized Partial
Sum Processes in the $M_{1}$ topology. We use the main results established in
Section 2 of the paper applied to the case of the self-normalized partial sum
process to obtain a weak convergence result that holds in the $M_{1}$ topology
under general conditions. Consider the function $\mathsf{g}\colon
D([0,1],\mathbb{R}^{2})\to\mathbb{R}^{2}$, defined by $\mathsf{g}(x)=x(1)$. It
is continuous with respect to the weak $M_{1}$ topology on
$D([0,1],\mathbb{R}^{2})$ (see Theorem 12.5.2 in Whitt Whitt, (2002)).
Therefore under the conditions given by Theorem 1, the continuous mapping
theorem yields $\mathsf{g}(L_{n})\xrightarrow{d}\mathsf{g}(L)$ as
$n\to\infty$, which implies that

(3.31)
$\displaystyle\bigg{(}\sum_{i=1}^{n}\frac{X_{i}}{a_{n}},\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}\bigg{)}\xrightarrow{d}\bigg{(}\sum_{i}\sum_{j}P_{i}\eta_{ij},\sum_{i}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\bigg{)}.$

In particular, it follows that

(3.32)
$\displaystyle\frac{V_{n}}{a_{n}^{2}}=\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}\xrightarrow{d}\sum_{i}\sum_{j}P_{i}^{2}\eta_{ij}^{2},$

where the limiting random variable has a stable distribution.

######  Theorem 3.

Let $(X_{n})_{n\in\mathbb{Z}}$ be a stationary sequence of random variables,
jointly regularly varying with index $\alpha\in(0,1)$, and of which the tail
process $(Y_{i})_{i\in\mathbb{Z}}$ almost surely has no two values of the
opposite sign. Suppose that Conditions 1 and 2 hold. Then

(3.33) $\displaystyle\frac{S_{\lfloor
n\,\cdot\rfloor}}{V_{n}}\xrightarrow{d}\frac{L_{1}(\,\cdot\,)}{\sqrt{L_{2}(1)}}\qquad\textrm{as}\
n\to\infty,$

in $D([0,1],\mathbb{R})$ endowed with the $M_{1}$ topology, where
$L=(L_{1},L_{2})$ as given in Theorem 1.

###### Proof.

The proof of Theorem 3 is shown in Appendix B of the paper. ∎

######  Theorem 4.

Let $(X_{n})_{n\in\mathbb{Z}}$ be a stationary sequence of random variables,
jointly regularly varying with index $\alpha\in[1,2)$, and of which the tail
process $(Y_{i})_{i\in\mathbb{Z}}$ almost surely has no two values of the
opposite sign. Suppose that Conditions 1 and 2 hold. Then

(3.34) $\displaystyle\frac{S_{\lfloor
n\,\cdot\rfloor}}{V_{n}}\xrightarrow{d}\frac{L_{1}(\,\cdot\,)}{\sqrt{L_{2}(1)}}\qquad\textrm{as}\
n\to\infty,$

in $D([0,1],\mathbb{R})$ endowed with the $M_{1}$ topology, where
$L=(L_{1},L_{2})$ as given in Theorem 1.

###### Proof.

The proof of Theorem 4 is shown in Appendix B of the paper. ∎

Both limit processes $L_{1}$ and $L_{2}$ denote independent strictly
$\alpha-$stable and $\alpha/2-$stable random variables respectively. Thus, our
aim is to prove that the limit $L_{1}(\,\cdot\,)\big{/}\sqrt{L_{2}(1)}$, that
joint weak convergence induces in Theorem 3 and 4, gives an $\alpha-$stable
random variable.

##  Main Applications

In this section, we establish the weak convergence of joint functionals to the
space $D[(0,1)]$ equipped with the $M_{1}$ topology for various processes. Our
aim is to provide applications in which the finite-dimensional distributions
of self-normalized partial-sum processes converge to those of a strictly
$\alpha-$stable process and the partial-sum process is tight. We focus on
demonstrating that the weak convergence of self-normalized processes of joint
functionals of random variables induced by time series models hold to the
$M_{1}$ topology based on our theoretical results. Thus, weak convergence
arguments are essential for determining the limit behaviour of estimators and
related test statistics that hold for sequences with a regular variation index
$\alpha\in(0,2)$. In other words, we focus on establishing the weak
convergence of the joint functional $L_{n}(t)$:

(4.1) $\displaystyle
L_{n}(t)=\left(\sum_{k=1}^{[nt]}\frac{X_{k}}{a_{n}},\sum_{k=1}^{\lfloor
nt\rfloor}\frac{X_{k}^{2}}{a_{n}^{2}}\right),\qquad t\in[0,1],$

for $\alpha\in(0,1)$ or $\alpha\in[1,2)$, where the stochastic process
$\left(X_{k}\right)_{k\in\mathbb{N}}$ corresponds to the stationary, regularly
varying sequence induced from a regression model. The main idea here is to
show that these results are consistent with the results provided by BKS for
processes from a regularly varying distribution with index $\alpha\in(0,2)$
when one considers the weak convergence of the self-normalized partial sum
functionals with appropriate centering and normalizing sequences.

We focus on time series processes for which the weak convergence of self-
normalized partial sum processes do not converge in the standard $J_{1}$
topology. The particular weak convergence result is strong in the sense that
it holds regardless whether clustering of only values of the same sign occurs
in the limit process. In other words, the $J_{1}$ topology is appropriate when
large values of $X_{n}$ do not cluster, while in the contrary existence of
clusters of large values implies that $J_{1}$ convergence fail to hold (see,
Basrak et al., (2010), Krizmanić, (2016)). On the other hand, when considering
the joint weak convergence of functionals within the $M_{1}$ topology related
asymptotic theory results can be established. Therefore, we consider the weak
convergence of such joint functionals to the $M_{1}$ topology, which is more
appropriate in the presence of clusters of extreme values. Although, we have
not mention any related mixing conditions, we mainly focus in the case of weak
dependence and joint regular variation for the sequence
$\left(X_{k}\right)_{k\in\mathbb{N}}$.

######  Example 1 (Linear process).

Let $\left(Z_{i}\right)_{i\in\mathbb{Z}}$ be an i.i.d regularly varying
sequence with index $\alpha\in(0,2)$. Consider the following linear process
for the i.i.d sequence $\left(X_{i}\right)_{i\in\mathbb{Z}}$ such that

(4.2) $\displaystyle X_{i}=\sum_{j=0}^{\infty}\varphi_{j}Z_{i-j},\ \ \ \
k\in\mathbb{Z},$

where the sequence of real numbers $(\varphi_{j})$ is such that

(4.3)
$\displaystyle\sum_{j=0}^{\infty}\left|\varphi_{j}\right|^{\beta}<\infty\ \ \
\ \text{for some}\ \ 0<\beta<\alpha,\ \beta\leq 1,\ \ \ \text{such that}\ \ \
\sum_{j=0}^{\infty}\varphi_{j}\neq 0.$

Moreover, it holds that

(4.4)
$\displaystyle\underset{x\to\infty}{\mathsf{lim}}\frac{\mathbb{P}\left(|X_{0}|>x\right)}{\mathbb{P}\left(|Z_{0}|>x\right)}=\sum_{j=0}^{\infty}\left|\varphi_{j}\right|^{\alpha},\
\ \alpha\in(0,2).$

Then, the tail process $\left(Y_{i}\right)_{k\in\mathbb{N}}$ of the linear
process $\left(X_{i}\right)$ is of the following form:

(4.5) $\displaystyle
Y_{i}=\frac{\varphi_{n+M}}{\left|\varphi_{M}\right|}|Y_{0}|\Theta_{i}^{Z_{i}},\
\ \ n\in\mathbb{Z},$

where $J$ is an $\left\\{-1,1\right\\}$-valued random variable such that
$\mathbb{P}\left(\Theta_{i}^{Z_{i}}=1\right)=p$ and
$\mathbb{P}\left(\Theta_{i}^{Z_{i}}=0\right)=q$ for all
$i\in\left\\{1,..,n\right\\}$ and $M$ is an integer valued random variable
with probability distribution

(4.6)
$\displaystyle\mathbb{P}\big{(}M=m\big{)}=\frac{|\varphi_{j}|^{\alpha}}{\sum_{i=0}^{\infty}|\varphi_{j}|^{\alpha}},\
m\geq 0.$

All the above assumptions ensure that the tail process has no two values of
the opposite sign, which ensures that the theoretical conditions of our
framework are satisfied. Thus, if we consider the corresponding finite order
linear process, we can apply the main theory since in this case all conditions
of our theorems are satisfied.

Hence, for the truncated linear process:
$X_{i}=\sum_{j=0}^{m}\varphi_{j}Z_{i-j}$ it holds that,

(4.7) $\displaystyle
L_{1}(\,\cdot\,)=\begin{cases}\displaystyle\frac{\sum_{j=0}^{m}\varphi_{j}}{\kappa}\sum_{T_{i}\leq.}P_{i}\Theta_{i}^{Z_{i}},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\frac{1}{\kappa}\sum_{T_{i}\leq.}P_{i}\Theta_{i}^{Z_{i}}\sum_{j=0}^{m}\varphi_{j}\mathbf{1}_{\left\\{P_{i}\left|\varphi_{j}\right|/\kappa\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

where $L_{1n}(t):=\frac{1}{a_{n}}\sum_{i=1}^{\lfloor nt\rfloor}X_{i}$ and
$X_{i}=\sum_{j=0}^{m}\varphi_{j}Z_{i-j}$.

Therefore, we define with

(4.8) $\displaystyle
L_{2}(\,\cdot\,)=\begin{cases}\displaystyle\frac{\sum_{j=0}^{m}\varphi_{j}}{\kappa}\sum_{T_{i}\leq.}P^{2}_{i}\left(\Theta_{i}^{Z_{i}}\right)^{2},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\frac{1}{\kappa}\sum_{T_{i}\leq.}P^{2}_{i}\left(\Theta_{i}^{Z_{i}}\right)^{2}\sum_{j=0}^{m}\varphi_{j}\mathbf{1}_{\left\\{P^{2}_{i}\left|\varphi_{j}\right|/\kappa\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x^{2}\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

where $L_{2n}(t):=\frac{1}{a^{2}_{n}}\sum_{i=1}^{\lfloor nt\rfloor}X^{2}_{i}$
and $X^{2}_{i}=\left(\sum_{j=0}^{m}\varphi_{j}Z_{i-j}\right)^{2}$.

Then, based on Theorem 3 and Theorem 4, it holds that

$\displaystyle\frac{L_{1n}(t)}{L_{2n}(1)}$
$\displaystyle\overset{\mathsf{def}}{=}\frac{\displaystyle\frac{1}{a_{n}}\sum_{i=1}^{\lfloor
nt\rfloor}X_{i}}{\displaystyle\frac{1}{a^{2}_{n}}\sum_{i=1}^{n}X^{2}_{i}}=\frac{\displaystyle\frac{1}{a_{n}}\sum_{i=1}^{\lfloor
nt\rfloor}\sum_{j=0}^{m}\varphi_{j}Z_{i-j}}{\displaystyle\frac{1}{a^{2}_{n}}\sum_{i=1}^{n}\left(\sum_{j=0}^{m}\varphi_{j}Z_{i-j}\right)^{2}}$

$\displaystyle\xrightarrow{d}\begin{cases}\frac{\displaystyle\frac{\sum_{j=0}^{m}\varphi_{j}}{\kappa}\sum_{T_{i}\leq.}P_{i}\Theta_{i}^{Z_{i}}}{\displaystyle\frac{\sum_{j=0}^{m}\varphi_{j}}{\kappa}\sum_{T_{i}\leq.}P^{2}_{i}\left(\Theta_{i}^{Z_{i}}\right)^{2}},&\
\text{if}\ \alpha\in(0,1)\\\ \\\ \frac{\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\frac{1}{\kappa}\sum_{T_{i}\leq.}P_{i}\Theta_{i}^{Z_{i}}\sum_{j=0}^{m}\varphi_{j}\mathbf{1}_{\left\\{P_{i}\left|\varphi_{j}\right|/\kappa\right\\}}-t\int_{u<\left|x\right|\leq
1}x\mu(dx)\right)}{\left\\{\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\frac{1}{\kappa}\sum_{T_{i}\leq.}P^{2}_{i}\left(\Theta_{i}^{Z_{i}}\right)^{2}\sum_{j=0}^{m}\varphi_{j}\mathbf{1}_{\left\\{P^{2}_{i}\left|\varphi_{j}\right|/\kappa\right\\}}-n\int_{u<\left|x\right|\leq
1}x^{2}\mu(dx)\right)\right\\}^{1/2}},&\ \text{if}\ \alpha\in[1,2)\end{cases}$
$\displaystyle\equiv\frac{L_{1}(\,\cdot\,)}{\sqrt{L_{2}(1)}},\qquad\textrm{as}\
n\to\infty.$

######  Remark 10.

Our first application considers a linear process which is a standard
application in the literature (see also Basrak et al., (2010)). Although, a
linear process is considered a special case in which the $\beta-$mixing
condition doesn’t hold, our fundamental framework which corresponds to more
general dependence structures encompasses the case of linear processes.
Specifically, for this example, we focus on implementing our main results in
order to establish the weak convergence of self-normalized partial sum
processes of joint functionals equipped with the $M_{1}$ topology. On the
other hand, in time series regression settings asymptotic independence
conditions such as mixing conditions are usually proposed to replace
independence, among which $\beta-$mixing is an important dependent structure
and has been connected with a large class of time series models including ARMA
models and GARCH models.

######  Example 2 (Garch(1,1) process).

Consider the standard Garch model (see, Bollerslev, (1986))

(4.9) $\displaystyle X_{k}=\sigma_{k}Z_{k},\ \ \ k\in\mathbb{Z}.$

where $\left(Z_{k}\right)_{k\in\mathbb{N}}$ is a sequence of i.i.d random
variables with $\mathbb{E}(Z_{1})=0$ and $\mathbb{E}(Z^{2}_{1})=1$ such that

(4.10)
$\displaystyle\sigma_{k}^{2}=\omega+\left(a_{1}Z_{k-1}^{2}+b_{1}\right)\sigma_{k-1}^{2},\
\ \ k\in\mathbb{Z}.$

where $a_{1}$ and $b_{1}$ are non-negative parameters ensuring the non-
negativity of the squared volatility process $(\sigma_{k}^{2})$. Moreover, we
assume that $Z_{1}$ is symmetric, has a positive Levesgue density on
$\mathbb{R}$ and there exists $\alpha\in(0,2)$ such that

(4.11)
$\displaystyle\mathbb{E}\big{[}\left(a_{1}Z_{1}^{2}+b_{1}\right)^{\alpha}\big{]}=1\
\ \ \text{and}\ \ \
\mathbb{E}\big{[}\left(a_{1}Z_{1}^{2}+\beta_{1}\right)^{\alpha}\mathsf{ln}\left(a_{1}Z_{1}^{2}+b_{1}\right)\big{]}<\infty.$

In other words, based on the well-known property that under fairly weak
conditions on the distributions of the noise $(Z_{t})$, its finite-dimensional
distributions are regularly varying (see, mikosch2002whittle) and thus
satisfies the main conditions we impose in our framework.

Specifically, it is known that the processes $\left(\sigma_{k}^{2}\right)$ and
$\left(X_{k}^{2}\right)$ are regularly varying with index $\alpha$ and
strongly mixing geometric rate. In particular,

(4.12)
$\displaystyle\mathbb{P}\left(X_{1}^{2}>x\right)\sim\mathbb{E}\left|Z_{1}\right|^{2\alpha}\mathbb{P}\left(\sigma_{1}^{2}>x\right)\sim\mathbb{E}\left|Z_{1}\right|^{2\alpha}c_{1}x^{-\alpha},\
\ \text{as}\ \ x\to\infty.$

######  Remark 11.

Garch(1,1) processes are stochastic volatility models used in the statistics
and econometrics literature for modelling conditional volatility. Garch
processes have a main advantage to Arch-type processes in capturing clusters
of volatility or clusters of extreme values. Moreover, heavy-tailed Garch
processes are found to be robust to capture volatility clustering effects in
the literature. Therefore, in order for the conditions we impose in our
framework to be satisfied, for the remaining derivations of this example, we
consider partial sum processes of squared Garch(1,1) processes.

######  Example 3.

Consider the squared GARCH(1,1) process can be embedded in the 2-dimensional
stochastic recurrence equation (SRE) such that

(4.13)
$\displaystyle\boldsymbol{X}_{k}=\boldsymbol{A}_{k}\boldsymbol{X}_{k-1}+\boldsymbol{E}_{k}$

where

(4.14) $\displaystyle\boldsymbol{X}_{k}=\begin{pmatrix}X_{k}^{2}\\\
\sigma_{k}^{2}\end{pmatrix},\ \ \
\boldsymbol{A}_{k}=\begin{pmatrix}a_{1}Z_{k}^{2}\ &\ b_{1}Z_{k}^{2}\\\ a_{1}\
&\ b_{1}\end{pmatrix},\ \ \ \boldsymbol{X}_{k}=\begin{pmatrix}\omega
Z_{k}^{2}\\\ \omega\end{pmatrix}.$

Since the process $\left(X_{k}\right)$ is nonnegative, its tail process cannot
have two values of the opposite sign. Moreover, the conditional volatility
sequence $(\sigma_{k}^{2})_{k\in\mathbb{Z}}$ has all elements of its time
series observations being positive. Consequently, the vector process
$\boldsymbol{X}_{k}$ satisfies the no sign switching condition (see, Basrak et
al., (2010)). Thus, we consider the weak convergence of the partial sum
process $V_{n}(.)$ for the cases when $\alpha\in(0,1)$ and when
$\alpha\in[1,2)$.

The partial sum process $\boldsymbol{L}_{1n}:=V_{n}(.)$ is defined as below

(4.15) $\displaystyle\boldsymbol{V}_{n}(t)=\sum_{k=1}^{\lfloor
nt\rfloor}\frac{\boldsymbol{X}_{k}^{2}}{\alpha_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{\boldsymbol{X}_{1}^{2}}{\alpha_{n}}\mathbf{1}_{\left\\{\frac{\boldsymbol{X}_{1}^{2}}{\alpha_{n}}\leq
1\right\\}}\right],\ t\in[0,1]$

Note that the partial sum process $\boldsymbol{V}_{n}(t)$ given by expression
(4.17) converges in distribution to an $\alpha-$stable Lévy process $V(.)$,
with a characteristic triple $\left(0,\nu,b\right)$, in $D[0,1]$ equipped with
the $M_{1}$ topology since it satisfies the Assumptions of the main theory of
the paper. Moreover, since
$\boldsymbol{X}_{k}=\big{(}X_{k}^{2},\sigma^{2}_{k}\big{)}^{\prime}$ is a
vector stochastic process, for $k\in\left\\{1,...,n\right\\}$, the partial sum
process $\boldsymbol{V}_{n}(t)$ can be expressed in the following matrix form

(4.16)
$\displaystyle\boldsymbol{L}_{1n}:=\boldsymbol{V}_{n}(t)=\begin{pmatrix}\displaystyle\frac{1}{a_{n}}\sum_{k=1}^{\lfloor
nt\rfloor}X_{k}^{2}\\\ \\\ \displaystyle\frac{1}{a_{n}}\sum_{k=1}^{\lfloor
nt\rfloor}\sigma_{k}^{2}\end{pmatrix}-\lfloor
nt\rfloor\begin{pmatrix}\displaystyle\frac{1}{a_{n}}\mathbb{E}\left[X_{1}^{2}\mathbf{1}_{\left\\{\frac{\boldsymbol{X}_{1}^{2}}{\alpha_{n}}\leq
1\right\\}}\right]\\\ \\\
\displaystyle\frac{1}{a_{n}}\mathbb{E}\left[\sigma_{1}^{2}\mathbf{1}_{\left\\{\frac{\sigma_{1}^{2}}{\alpha_{n}}\leq
1\right\\}}\right]\end{pmatrix},\ t\in[0,1]$

Similarly, we define the partial-sum process $\boldsymbol{W}_{n}(t)$ as below

(4.17)
$\displaystyle\boldsymbol{L}_{2n}:=\boldsymbol{W}_{n}(t)=\begin{pmatrix}\displaystyle\frac{1}{a^{2}_{n}}\sum_{k=1}^{\lfloor
nt\rfloor}X_{k}^{4}\\\ \\\ \displaystyle\frac{1}{a^{2}_{n}}\sum_{k=1}^{\lfloor
nt\rfloor}\sigma_{k}^{4}\end{pmatrix}-\lfloor
nt\rfloor\begin{pmatrix}\displaystyle\frac{1}{a^{2}_{n}}\mathbb{E}\left[X_{1}^{4}\mathbf{1}_{\left\\{\frac{\boldsymbol{X}_{1}^{4}}{\alpha_{n}}\leq
1\right\\}}\right]\\\ \\\
\displaystyle\frac{1}{a^{2}_{n}}\mathbb{E}\left[\sigma_{1}^{4}\mathbf{1}_{\left\\{\frac{\sigma_{1}^{4}}{\alpha_{n}}\leq
1\right\\}}\right]\end{pmatrix},\ t\in[0,1]$

Furthermore, by applying the main results of the paper we can consider the
joint functional convergence of the corresponding self-normalized partial sum
process such that

(4.18) $\displaystyle\frac{\boldsymbol{L}_{1\lfloor
nt\rfloor}}{\boldsymbol{L}_{2n}(\,t\,)}\xrightarrow{d}\frac{\boldsymbol{L}_{1}(\,t\,)}{\sqrt{\boldsymbol{L}_{2}(\,1\,)}}\qquad\textrm{as}\
n\to\infty.$

######  Remark 12.

The assumption that the tail process has no two values of the opposite sign is
crucial to obtain weak convergence of the partial sum process in the $M_{1}$
topology. One of the restrictions of the particular topology is that it can
handle several instantaneous jumps only if they are in the same direction. On
the other hand, there appear to be some ways of omitting the no sign switching
condition. A possibility is to avoid the within-cluster fluctuations in the
partial sum process, by smoothing out its trajectories or by considering the
process $t\mapsto S_{r_{n}\lfloor k_{n}t\rfloor}$, which then implies that
convergence holds in the stronger $J_{1}$ topology. Our main results build on
the framework of BKS to establish the weak convergence of self-normalized
partial sum processes within the more restrictive $M_{1}$ topology (see,
Krizmanić, (2016, 2018, 2020)).

Therefore, the above illustrate theoretical examples show cases in which the
weak convergence of the partial-sum process, using its corresponding
characteristic function representation, holds in the $D[0,1]$ space equipped
with the $M_{1}$ topology. Similarly, one can prove that the second coordinate
of the functional weakly converges to the $M_{1}$ topology which allows us to
consider the joint weak convergence of the self-normalized partial-sum
processes under investigation. To do this, we need to determine the exact
values of the characteristic triple that correspond to both coordinates of the
joint functional process. Specifically, for cases in which the standard
$J_{1}$ topology fails to apply, we are interested to establish joint weak
convergence in the $M_{1}$ topology which although is more restrictive it
allows to obtain asymptotic results for extreme models and cluster-based
processes which are more likely to have underline regular varying processes
that satisfy the aforementioned theoretical results derived in this paper.

##  Conclusion

According to Basrak et al., (2010), for a stationary, regularly varying
sequence for which clusters of high-thredshold excesses can be broken down
into asymptotically independent blocks, the properly centered partial sum
process $\left(V_{n}(t)\right)_{t\in[0,1]}$ converges to an $\alpha-$stable
Lévy process in the space $D[0,1]$ endowed with Skorokhod’s $M_{1}$ metric
under the condition that all extremes within one such cluster have the same
sign. Therefore, the main proofs and applications in this paper are extending
the particular conjecture to the case of self-normalized partial sum
processes.

In general, the $M_{1}$ topology is considered to be weaker than the more
commonly used Skorohod $J_{1}$ topology, the $M_{1}$ topology allows to
consider the case in which weak convergence has to be established under the
presence of clustering of extreme values. This particular effect although is
examined from the perspective of stationary stochastic processes, interesting
questions arise in the case of nonstationary time series processes under the
presence of clustering. Therefore, further research includes extensions to
nonstationary sequences which is a special case. In that case, one would need
to first of all check that the corresponding conditions and definitions (e.g.,
tail index of the sequence etc), hold and then prove the general case of the
theorem. Then and iff the particular theorem holds we can apply the equivalent
representation of self-normalized partial sum processes to nonstationary
sequences without worrying that the weak convergence argument will not hold.
Further interesting extensions of our work include among other cases such as
when one considers joint weak convergence of empirical processes of cluster
functionals (see, Engelke et al., (2022) and Drees and Rootzen, (2010)).

#### Acknowledgements

I am grateful and indebted to Associate Professor Danijel Krizmanic from the
Department of Mathematics, University of Rijeka for providing guidance
throughout the development of this research project. Moreover, he provided the
main steps for the proof in the case where $\alpha\in(0,1)$ which helped me
establish the case $\alpha\in[1,2)$.

##  Appendix A Proofs of Main Results in Section 1

###  Proof of Lemma 1

###### Proof.

Since a continuous function on a compact set is bounded and attains its
minimum value it holds that

(A.1) $\displaystyle
C^{\uparrow}_{0}([0,1],\mathbb{R})=\bigcup_{k=1}^{\infty}C^{\uparrow}_{k}([0,1],\mathbb{R}),$

where $C^{\uparrow}_{k}([0,1],\mathbb{R})$ is the subset of functions $x$ in
$C^{\uparrow}_{0}([0,1],\mathbb{R})$ with $x(0)\geq 1/k$.

Consider $(x_{n})_{n}$ to be a sequence in
$C^{\uparrow}_{k}([0,1],\mathbb{R})$ which converges to $x$ with respect to
the $J_{1}$ topology. Since a discontinuous element of $D([0,1],\mathbb{R})$
cannot be approximated in the $J_{1}$ topology by a sequence of continuous
functions, the limit $x$ must be continuous and the convergence is uniform
(see for example Kulik and Soulier, (2020), Proposition C.2.4).

Thus, for $0\leq s<t\leq 1$, from $x_{n}(s)\leq x_{n}(t)$ letting $n\to\infty$
we obtain $x(s)\leq x(t)$, i.e. $x$ is nondecreasing. Similarly we obtain
$x(0)\geq 1/k$, which means that $x\in C^{\uparrow}_{k}([0,1],\mathbb{R})$.
Hence $C^{\uparrow}_{k}([0,1],\mathbb{R})$ is a closed, and then also a
measurable subset of $D([0,1],\mathbb{R})$ with the $J_{1}$ topology.

Since the Borel $\sigma$–fields associated with the non-uniform Skorokhod
topologies all coincide with the Kolmogorov $\sigma$–field generated by the
projection maps, the set $C^{\uparrow}_{k}([0,1],\mathbb{R})$ is measurable
with the $M_{1}$ topology for each $k$, and therefore the same holds for
$C^{\uparrow}_{0}([0,1],\mathbb{R})$. ∎

###  Proof of Lemma 3

###### Proof.

Consider an arbitrary pair $(x,y)\in D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})$.

Suppose that $d_{p}((x_{n},y_{n}),(x,y))\to 0$ as $n\to\infty$. We need to
show that $h(x_{n},y_{n})\to h(x,y)$ in the $M_{1}$ topology, that is,
$d_{M_{1}}(x_{n}/y_{n},x/y)\to 0$. By the definition of the metric $d_{p}$, we
have $d_{M_{1}}(x_{n},x)\to 0$ and $d_{M_{1}}(y_{n},y)\to 0$ as $n\to\infty$.
Since for monotone functions the $M_{1}$ convergence is equivalent to
pointwise convergence in a dense subset $A$ of $[0,1]$ including $0$ and $1$
(Whitt, (2002), Corollary 12.5.1), the convergence $d_{M_{1}}(y_{n},y)\to 0$,
which implies that

(A.2) $\displaystyle\frac{1}{y_{n}(t)}\to\frac{1}{y(t)},\qquad t\in A.$

Furthermore, since the functions $1/y,1/y_{n}$ ($n\in\mathbb{N}$) are also
monotone, $d_{M_{1}}(1/y_{n},1/y)\to 0$ as $n\to\infty$. Noting that the
function $1/y$ is continuous, we can apply Lemma 3 to conclude that
$d_{M_{1}}(x_{n}/y_{n},x/y)\to 0$ as $n\to\infty$. ∎

##  Appendix B Proofs of Main Results in Section 2

###  Proof of Lemma 5

###### Proof.

Take an arbitrary $\eta\in\Lambda$ and suppose that
$\eta_{n}\xrightarrow{v}\eta$ in $\mathbf{M}_{p}([0,1]\times\mathbb{E})$. We
need to show that $\Phi^{(u)}(\eta_{n})\to\Phi^{(u)}(\eta)$ in
$D([0,1],\mathbb{R}^{2})$ according to the weak $M_{1}$ topology. By Theorem
12.5.2 in Whitt, (2002), it suffices to prove that, as $n\to\infty$,

(B.1) $\displaystyle
d_{p}\left(\Phi^{(u)}(\eta_{n}),\Phi^{(u)}(\eta)\right)=\max_{k=1,2}d_{M_{1}}\left(\Phi^{(u)}_{k}(\eta_{n}),\Phi^{(u)}_{k}(\eta)\right)\to
0.$

Following, with small modifications, the proof of Lemma 3.2 in Basrak et al.,
(2010) for each coordinate separately, we obtain
$d_{M_{1}}\left(\Phi^{(u)}_{1}(\eta_{n}),\Phi^{(u)}_{1}(\eta)\right)\to 0$ and
$d_{M_{1}}\left(\Phi^{(u)}_{2}(\eta_{n}),\Phi^{(u)}_{2}(\eta)\right)\to 0$ as
$n\to\infty$. Therefore we conclude that $\Phi^{(u)}$ is continuous at $\eta$.
∎

###  Proof of Theorem 1

The following proof is based on the manuscript of Danijel Krijmanic who kindly
shared. Related references to the main steps as presented in related studies
in the literature are also carefully explained. Thus, in a similar manner as
in Krizmanić, (2020) who establish the joint weak convergence of partial sum
and maxima processes, the following steps are adjusted accordingly in order to
accommodate the fact that we are interested in establishing the joint weak
convergence of the stochastic process $X_{t}$ and its square $X_{t}^{2}$
rather the corresponding maxima of this process. Consequently, the second
coordinate of expression (B.2) corresponds to the squared term (e.g., see
proof of Theorem 3.4 in Krizmanić, (2020)). Take an arbitrary $u>0$, and
consider

(B.2) $\displaystyle\Phi^{(u)}(N_{n})(t)=\left(\sum_{i/n\leq
t}\frac{X_{i}}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}>u\big{\\}}},\sum_{i/n\leq
t}\frac{X_{i}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X^{2}_{i}|}{a^{2}_{n}}>u\big{\\}}}\right),\qquad
t\in[0,1].$

From Lemma 4 and Lemma 5 we know that $\Phi^{(u)}$ is continuous on the set
$\Lambda$ defined by (3.5). Recall that this set is basically an extension of
the space $E$ as defined in Section 4.2 of Basrak et al., (2018), in order to
accommodate the more general setting we have in this study. This set almost
surely contains the limiting point process $N$ such that
$N_{n}\xrightarrow{d}N:=\sum_{i}\sum_{j}\delta_{(T_{i},P_{i}\eta_{ij})}\in[0,1]\times\mathbb{E}\
\ \ \text{as}\ n\to\infty$.

Hence an application of the continuous mapping theorem yields that
$\Phi^{(u)}(N_{n})\xrightarrow{d}\Phi^{(u)}(N)$ in $D([0,1],\mathbb{R}^{2})$
under the weak $M_{1}$ topology (see, similar argumentation in Krizmanić,
(2018, 2020). Therefore, we can establish the convergence of the functional
$L_{n}^{(u)}(\,\cdot\,)$ as below

(B.3) $L_{n}^{(u)}(\,\cdot\,):=\left(\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}}{a_{n}}\boldsymbol{1}_{\bigl{\\{}\frac{|X_{i}|}{a_{n}}>u\bigr{\\}}},\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\bigl{\\{}\frac{|X^{2}_{i}|}{a^{2}_{n}}>u\bigr{\\}}}\right)\\\
\xrightarrow{d}L^{(u)}(\,\cdot\,):=\left(\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}},\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}\right).$

where $\left\\{a_{n}\right\\}$ is a sequence of norming constants which
ensures that convergence in distribution holds as $u\to 0$.

Moreover, let $U_{i}=\sum_{j}\eta_{ij}$ and
$\widetilde{U}_{i}=\sum_{j}\eta_{ij}^{2}$, where
$i\in\left\\{1,2,\ldots\right\\}$. Then, since in Theorem 1 we assume that
$\alpha<1$, from the moment condition given in Corollary 1 (2.10) (e.g., see
Davis and Hsing, (1995)), the following conditions apply for the two
coordinates of the functional under consideration.

$\displaystyle\mathrm{E}|U_{1}|^{\alpha}$
$\displaystyle\leq\mathrm{E}\Big{(}\sum_{j}|\eta_{j}|\Big{)}^{\alpha}<\infty,$
$\displaystyle\mathrm{E}|\widetilde{U}_{1}|^{\alpha/2}=\mathrm{E}\Big{(}\sum_{j}|\eta_{j}|^{2}\Big{)}^{\alpha/2}$
$\displaystyle\leq\mathrm{E}\Big{(}\sum_{j}|\eta_{j}|\Big{)}^{\alpha}<\infty,$

where in the last relation we used also the triangle inequality
$|\sum_{i}a_{i}|^{s}\leq\sum_{i}|a_{i}|^{s}$ with $s\in(0,1]$.

Using Proposition 5.2 and Proposition 5.3 given in Resnick, (2007), it holds
that $P_{i}|U_{i}|$, for $i\in\left\\{1,2,\ldots\right\\}$, are the points of
a Poisson process with intensity measure
$\theta\alpha\mathbf{E}|U_{1}|^{\alpha}x^{-\alpha-1}\,\mathrm{d}x$, where
$x>0$. Since these points are summable it holds that

(B.4)
$\displaystyle\sum_{i=1}^{\infty}P_{i}|U_{i}|\leq\sum_{i=1}^{\infty}\sum_{j}P_{i}|\eta_{ij}|<\infty$

almost surely (see the proof of Theorem 3.1 in Davis and Hsing, (1995)). Thus
for all $t\in[0,1]$

(B.5)
$\displaystyle\sum_{T_{i}\leq.}^{t}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}\to\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}$

almost surely as $u\to 0$. In other words, expression (B.5) provides a
summability condition that ensures weak convergence of a partial sum process
to Lévy processes (e.g., see Krizmanić, (2020)).

Furthermore, the first summand has a lower bound $\left\\{T_{i}\leq.\right\\}$
and an upper bound $t$ which implies that we are summing up all points up to
until time $T_{i}$.

By the dominated convergence theorem

(B.6) $\displaystyle\sup_{t\in[0,1]}\bigg{|}\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}-\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}\bigg{|}\leq\sum_{i=1}^{\infty}\sum_{j}P_{i}|\eta_{ij}|\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|\leq
u\\}}\to 0$

almost surely as $u\to 0$.

Since uniform convergence implies Skorokhod $M_{1}$ convergence, we get

(B.7)
$d_{M_{1}}\left(\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}},\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\right)\to
0$

almost surely as $u\to 0$.

Similarly, $P_{i}^{2}\widetilde{U}_{i}$, $i\in\left\\{1,2,\ldots\right\\}$,
are the points of a Poisson process with intensity measure
$(\theta\alpha/2)\mathrm{E}(\widetilde{U}_{1})^{\alpha/2}x^{-\alpha/2-1}\,\mathrm{d}x$
for $x>0$, and it holds that

(B.8)
$d_{M_{1}}\left(\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}},\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\right)\to
0$

almost surely as $u\to 0$. Recalling the definition of the metric $d_{p}$ in
(2.16), from (B.7) and (B.8) we obtain that almost surely as $u\to 0$,

$d_{p}\left(L^{(u)},L\right)\to 0$

where

(B.9) $\displaystyle L(t)=\left(\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij},\sum_{T_{i}\leq
t}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\right).$

Since almost sure convergence implies weak convergence, we have, as $u\to 0$,

(B.10) $L^{(u)}\xrightarrow{d}L$

in $D([0,1],\mathbb{R}^{2})$ endowed with the weak $M_{1}$ topology. By
Proposition 5.2 and Proposition 5.3 in Resnick, (2007), the process

(B.11) $\displaystyle\sum_{i}\delta_{(T_{i},\sum_{j}P_{i}\eta_{ij}})$

is a Poisson process with intensity measure $Leb\times\nu_{1}$, and

(B.12) $\displaystyle\sum_{i}\delta_{(T_{i},\sum_{j}P_{i}^{2}\eta_{ij}^{2}})$

is a Poisson process with intensity measure $Leb\times\nu_{2}$.

Furthermore, by the Itô representation of the Lévy process (see Resnick,
(2007), pages 150–153) and Theorem 14.3 in Ken-Iti, (1999),

(B.13) $\displaystyle L_{1}(t)=\sum_{T_{i}\leq t}\sum_{j}P_{i}\eta_{ij},\qquad
t\in[0,1],$

is an $\alpha$–stable Lévy process with characteristic triple
$(0,\nu_{1},(c_{+}-c_{-})\theta\alpha/(1-\alpha))$, and similarly

(B.14) $\displaystyle L_{2}(t)=\sum_{T_{i}\leq
t}\sum_{j}P_{i}^{2}\eta_{ij}^{2},\qquad t\in[0,1],$

is an $\alpha/2$–stable Lévy process with characteristic triple
$(0,\nu_{2},\theta\alpha\mathrm{E}(\sum_{j}\eta_{j}^{2})^{\alpha/2}/(2-\alpha))$.

Specifically, notice that the functional $L_{1}$ is an $\alpha-$stable Lévy
process while the $L_{2}$ functional represents $\alpha/2$–stable Lévy
process. Furthermore, within the same probability space it is well-known that
the sum of two independent Lévy processes gives a stable Lévy process with
intensity measure the sum of the intensity measures of the two processes.

Therefore, if we show that

(B.15) $\displaystyle\lim_{u\to
0}\limsup_{n\to\infty}\mathbb{P}\left(d_{p}(L_{n},L_{n}^{(u)})>\epsilon\right)=0$

for any $\epsilon>0$, from (B.30) and (B.10) by a variant of Slutsky’s theorem
(see Theorem 3.5 in Resnick, (2007)) it will follow that
$L_{n}\xrightarrow{d}L$ as $n\to\infty$, in $D([0,1],\mathbb{R}^{2})$ with the
weak $M_{1}$ topology.

Since the metric $d_{p}$ on $D([0,1],\mathbb{R}^{2})$ is bounded above by the
uniform metric on $D([0,1],\mathbb{R}^{2})$ (Theorem 12.10.3 in Whitt,
(2002)), it suffices to show that

(B.16) $\displaystyle\lim_{u\to
0}\limsup_{n\to\infty}\mathbb{P}\biggl{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\biggr{)}=0.$

Using stationarity and Markov’s inequality we get the bound

(B.17)
$\displaystyle\mathbb{P}\bigg{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\bigg{)}$
$\displaystyle=$ $\displaystyle\ \mathbb{P}\bigg{(}\sup_{t\in[0,1]}\
\max\bigg{\\{}\bigg{|}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}\bigg{|},\bigg{|}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X^{2}_{i}|}{a^{2}_{n}}\leq
u\big{\\}}}\bigg{|}\bigg{\\}}>\epsilon\bigg{)}$ $\displaystyle\leq$
$\displaystyle\
\mathbb{P}\bigg{(}\sum_{i=1}^{n}\frac{|X_{i}|}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}>\epsilon\bigg{)}+\mathbb{P}\bigg{(}\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X^{2}_{i}|}{a^{2}_{n}}\leq
u\big{\\}}}>\epsilon\bigg{)}$ $\displaystyle\leq$ $\displaystyle\
\epsilon^{-1}n\mathbf{E}\bigg{(}\frac{|X_{1}|}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\bigg{)}+\epsilon^{-1}n\mathbf{E}\bigg{(}\frac{X_{1}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X^{2}_{1}|}{a^{2}_{n}}\leq
u\big{\\}}}\bigg{)}.$

For the first term on the right-hand side of (B.3) we have

(B.18) $\displaystyle
n\mathbf{E}\bigg{(}\frac{|X_{1}|}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X^{2}_{1}|}{a^{2}_{n}}\leq
u\big{\\}}}\bigg{)}=u\cdot
n\mathbb{P}(|X_{1}|>a_{n})\cdot\frac{\mathbb{P}(|X_{1}|>ua_{n})}{\mathbb{P}(|X_{1}|>a_{n})}\cdot\frac{\mathbf{E}(|X_{1}|\boldsymbol{1}_{\\{|X_{1}|\leq
ua_{n}\\}})}{ua_{n}\mathbb{P}(|X_{1}|>ua_{n})},$

and for the second

(B.19) $\displaystyle
n\mathbf{E}\bigg{(}\frac{X_{1}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\bigg{)}=u^{2}\cdot
n\mathbb{P}(|X_{1}|>a_{n})\cdot\frac{\mathbb{P}(|X_{1}|>ua_{n})}{\mathbb{P}(|X_{1}|>a_{n})}\cdot\frac{\mathbf{E}(|X_{1}|^{2}\boldsymbol{1}_{\\{|X_{1}|\leq
ua_{n}\\}})}{u^{2}a_{n}^{2}\mathbb{P}(|X_{1}|>ua_{n})},$

Since $X_{1}$ is a regularly varying random variable with index $\alpha$, it
follows immediately that

(B.20)
$\displaystyle\frac{\mathbb{P}(|X_{1}|>ua_{n})}{\mathbb{P}(|X_{1}|>a_{n})}\to
u^{-\alpha}$

as $n\to\infty$. By Karamata’s theorem

(B.21)
$\displaystyle\lim_{n\to\infty}\frac{\mathbf{E}(|X_{1}|\boldsymbol{1}_{\\{|X_{1}|\leq
ua_{n}\\}})}{ua_{n}\mathbb{P}(|X_{1}|>ua_{n})}=\frac{\alpha}{1-\alpha}\qquad\textrm{and}\qquad\lim_{n\to\infty}\frac{\mathbf{E}(|X_{1}|^{2}\boldsymbol{1}_{\\{|X_{1}|\leq
ua_{n}\\}})}{u^{2}a_{n}^{2}\mathbb{P}(|X_{1}|>ua_{n})}=\frac{\alpha}{2-\alpha}.$

Therefore, taking into account relation (2.3), we get

(B.22) $\displaystyle
n\mathbf{E}\bigg{(}\frac{|X_{1}|}{a_{n}}\boldsymbol{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\bigg{)}\to
u^{1-\alpha}\frac{\alpha}{1-\alpha}\qquad\textrm{and}\qquad
n\mathbf{E}\bigg{(}\frac{X_{1}^{2}}{a_{n}^{2}}\boldsymbol{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\bigg{)}\to u^{2-\alpha}\frac{\alpha}{2-\alpha}$

as $n\to\infty$. Therefore from (B.3) we obtain

(B.23)
$\displaystyle\limsup_{n\to\infty}\mathbb{P}\bigg{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\bigg{)}\leq\epsilon^{-1}\alpha
u^{1-\alpha}\Big{(}\frac{1}{1-\alpha}+\frac{u}{2-\alpha}\Big{)}.$

Letting $u\to 0$, since $1-\alpha>0$, we finally obtain

(B.24) $\displaystyle\lim_{u\to
0}\limsup_{n\to\infty}\mathbb{P}\bigg{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\bigg{)}=0.$

###  Proof of Theorem 2

Notice that it holds that $b_{1n}^{(u)}\to b_{1}$ and $b_{2n}^{(u)}\to b_{2}$
as $u\to 0$

(B.25) $\displaystyle b_{1n}^{(u)}$
$\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\mathbb{E}\left[\frac{X_{1}}{a_{n}}\mathbf{1}_{\left\\{{\color[rgb]{1,0,0}u<}\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right],&\alpha\in[1,2).\end{cases}$ (B.26) $\displaystyle
b_{2n}^{(u)}$ $\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\mathbb{E}\left[\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\left\\{{\color[rgb]{1,0,0}u<}\frac{X_{1}^{2}}{a_{n}}\leq
1\right\\}}\right],&\alpha\in[1,2).\end{cases}$ (B.27) $\displaystyle b_{1}$
$\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\displaystyle\int_{{\color[rgb]{1,0,0}0<}|x|\leq
1}x\mu(dx),&\alpha\in[1,2).\end{cases}$ (B.28) $\displaystyle b_{2}$
$\displaystyle:=\begin{cases}0,&\alpha\in(0,1),\\\
\displaystyle\int_{{\color[rgb]{1,0,0}0<}x^{2}\leq
1}x^{2}\mu(dx),&\alpha\in[1,2).\end{cases}$

In other words, in the case where the regularly varying index $\alpha\in[1,2)$
similar to the approach of Basrak et al., (2018) (e.g., see Section 4.2), we
consider the centering constant.

###### Proof.

Therefore, similar to the case $\alpha\in(0,1)$ we aim to prove the joint weak
convergence of the functional $L_{n}(t)$ of both its coordinates to the
corresponding limit process. To do this, we take an arbitrary $u>0$, and
consider

(B.29) $\displaystyle\Phi^{(u)}(N_{n})(t)=\left(\sum_{i/n\leq
t}\frac{X_{i}}{a_{n}}\mathbf{1}_{\left\\{\frac{|X_{1}|}{a_{n}}\leq
u\right\\}},\sum_{i/n\leq
t}\frac{X_{i}^{2}}{a_{n}^{2}}\mathbf{1}_{\left\\{\frac{|X_{1}|}{a_{n}}\leq
u\right\\}}\right),\qquad t\in[0,1].$

From Lemma 4 and Lemma 5 we know that $\Phi^{(u)}$ is continuous on the set
$\Lambda$ and this set almost surely contains the limiting point process $N$
from (2.9). Hence an application of the continuous mapping theorem yields that
$\Phi^{(u)}(N_{n})\xrightarrow{d}\Phi^{(u)}(N)$ in $D([0,1],\mathbb{R}^{2})$
under the weak $M_{1}$ topology. Therefore, we can establish the convergence
of the functional $L_{n}^{(u)}(\,\cdot\,)$ as below

(B.30) $L_{n}^{(u)}(\,\cdot\,):=\left(\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}}{a_{n}}-\lfloor n\,\cdot\,\rfloor
b_{1n}^{(u)},\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}^{2}}{a_{n}^{2}}-\lfloor n\,\cdot\,\rfloor
b_{2n}^{(u)}\right)\\\ \xrightarrow{d}L^{(u)}(\,\cdot\,):=\left(\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}-(\,\cdot\,)b_{1}\right\\},\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\boldsymbol{1}_{\\{P^{2}_{i}|\eta^{2}_{ij}|>u\\}}-(\,\cdot\,)b_{2}\right\\}\right).$

where $\left\\{a_{n}\right\\}$ is a sequence of norming constants which
ensures that convergence in distribution holds as $u\to 0$, which proves that
$L^{(u)}\overset{d}{\to}L$ in $\mathcal{D}([0,1],\mathbb{R}^{2})$ endowed with
the weak $M_{1}$ topology.

Notice that the above joint convergence results implies that for each
coordinate of the joint functional it holds that

(B.31) $\displaystyle\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}}{a_{n}}-\lfloor n\,\cdot\,\rfloor b_{1n}^{(u)}$
$\displaystyle\to\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}-(\,\cdot\,)b_{1}\right\\},\
\ \ \text{as}\ n\to\infty$ (B.32) $\displaystyle\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}^{2}}{a_{n}^{2}}-\lfloor n\,\cdot\,\rfloor
b_{2n}^{(u)}$ $\displaystyle\to\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\boldsymbol{1}_{\\{P^{2}_{i}|\eta^{2}_{ij}|>u\\}}-(\,\cdot\,)b_{2}\right\\},\
\ \ \text{as}\ n\to\infty$

or in other words,

(B.33) $\displaystyle\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X_{i}}{a_{n}}-\lfloor
n\,\cdot\,\rfloor\mathbb{E}\left[\frac{X_{1}}{a_{n}}\mathbf{1}_{\left\\{{\color[rgb]{1,0,0}u<}\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right]\to\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P_{i}\eta_{ij}\boldsymbol{1}_{\\{P_{i}|\eta_{ij}|>u\\}}-(\,\cdot\,)\displaystyle\int_{{\color[rgb]{1,0,0}u<}|x|\leq
1}x\mu(dx)\right\\},\ \text{for}\ \ \alpha\in[1,2).$

and

(B.34) $\displaystyle\sum_{i=1}^{\lfloor
n\,\cdot\,\rfloor}\frac{X^{2}_{i}}{a^{2}_{n}}-\lfloor
n\,\cdot\,\rfloor\mathbb{E}\left[\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\left\\{{\color[rgb]{1,0,0}u<}\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right]\to\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq\,\cdot}\sum_{j}P^{2}_{i}\eta^{2}_{ij}\boldsymbol{1}_{\\{P^{2}_{i}|\eta^{2}_{ij}|>u\\}}-(\,\cdot\,)\displaystyle\int_{{\color[rgb]{1,0,0}u<}x^{2}\leq
1}x^{2}\mu(dx)\right\\},\ \text{for}\ \ \alpha\in[1,2).$

Next, as before we need to show that

(B.35) $\displaystyle\lim_{u\to
0}\limsup_{n\to\infty}\Pr\biggl{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\biggr{)}=0.$

Using stationarity and Markov’s inequality we get the bound

$\displaystyle\Pr\bigg{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\bigg{)}$
$\displaystyle=\Pr\bigg{(}\sup_{t\in[0,1]}\
\max\bigg{\\{}\bigg{|}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}-\lfloor
nt\rfloor\mathbb{E}\left(\frac{X_{1}}{a_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\right)\bigg{|},\bigg{|}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}^{2}}{a^{2}_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}-\lfloor
nt\rfloor\mathbb{E}\left(\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{1}|}{a_{n}}\leq
u\big{\\}}}\right)\bigg{|}\bigg{\\}}>\epsilon\bigg{)}$
$\displaystyle=\Pr\bigg{(}\max_{0\leq k\leq n}\
\bigg{|}\sum_{i=1}^{k}\bigg{\\{}\frac{X_{i}}{a_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}-\mathbb{E}\left(\frac{X_{i}}{a_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}\right)\bigg{\\}}\bigg{|}>\epsilon\bigg{)}$ $\displaystyle\
+\Pr\bigg{(}\max_{0\leq k\leq n}\
\bigg{|}\sum_{i=1}^{k}\bigg{\\{}\frac{X^{2}_{i}}{a^{2}_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}-\mathbb{E}\left(\frac{X^{2}_{i}}{a^{2}_{n}}\mathbf{1}_{\big{\\{}\frac{|X_{i}|}{a_{n}}\leq
u\big{\\}}}\right)\bigg{\\}}\bigg{|}>\epsilon\bigg{)}.$

Therefore, from previous conditions it follows that

(B.36) $\displaystyle\lim_{u\to
0}\limsup_{n\to\infty}\Pr\biggl{(}\sup_{t\in[0,1]}\|L_{n}(t)-L_{n}^{(u)}(t)\|>\epsilon\biggr{)}=0.$

Lastly, it remains to show that the Lévy process $V$ has a characteristic
triple $\left(0,\nu^{\prime},\gamma\right)$, which will imply that the limit
random variable is equivalent to the case where $\alpha\in(0,1)$ presented in
the literature, related to weak convergence of partial sum processes to the
space $D[0,1]$ equipped with the $M_{1}$ topology (see, Basrak et al.,
(2010)). To prove the required result we consider the analytical expression of
the characteristic function of the $\alpha-$stable random variable $V(1)$.

The Lévy process $V$ is the weak limit in the sense of finite dimensional
distributions of the partial sum process $V_{n}$ which is related to the
canonical measure in Feller’s representation of an infinitely divisible
characteristic function (see, Feller, (1971)) which is also discussed in Davis
and Hsing, (1995). Therefore, in the following derivations we follow the
approach presented both by Basrak et al., (2018) and Krizmanić, (2020),
following the derivations given in Theorem 3.2 of Davis and Hsing, (1995). In
other words, we obtain the analytical expression of the partial sum process
$V_{n}$ in terms of its characteristic function.

Specifically, we denote with $V_{n}(t)$ the Lévy process of the first
coordinate as below

(B.37) $\displaystyle
V_{n}(t):=L_{1n}(t)=\begin{cases}\displaystyle\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}&,\text{if}\ \alpha\in(0,1)\\\ \\\
\displaystyle\sum_{i=1}^{\lfloor nt\rfloor}\frac{X_{i}}{a_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{X_{1}}{\alpha_{n}}\mathbf{1}_{\left\\{\frac{X_{1}}{\alpha_{n}}\leq
1\right\\}}\right]&,\text{if}\ \alpha\in[1,2).\end{cases}$

where

(B.38) $\displaystyle
L_{1}(\,\cdot\,)=\begin{cases}\displaystyle\sum_{T_{i}\leq.}\sum_{j}P_{i}\eta_{ij},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\sum_{T_{i}\leq.}\sum_{j}P_{i}\eta_{ij}1_{\left\\{P_{i}\left|\eta_{ij}\right|>u\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

Similarly, we denote with $W_{n}(t)$ the Lévy process of the second coordinate
as below

(B.39) $\displaystyle
W_{n}(t):=L_{2n}(t)=\begin{cases}\displaystyle\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X^{2}_{i}}{a_{n}}&,\text{if}\ \alpha\in(0,1)\\\ \\\
\displaystyle\sum_{i=1}^{\lfloor nt\rfloor}\frac{X^{2}_{i}}{a_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{X^{2}_{1}}{\alpha_{n}}\mathbf{1}_{\left\\{\frac{X^{2}_{1}}{\alpha_{n}}\leq
1\right\\}}\right]&,\text{if}\ \alpha\in[1,2).\end{cases}$

where

(B.40) $\displaystyle
L_{2}(\,\cdot\,)=\begin{cases}\displaystyle\sum_{T_{i}\leq.}\sum_{j}P^{2}_{i}\eta^{2}_{ij},&\alpha\in(0,1),\\\
\displaystyle\underset{u\to
0}{\mathsf{lim}}\left(\sum_{T_{i}\leq.}\sum_{j}P^{2}_{i}\eta^{2}_{ij}1_{\left\\{P^{2}_{i}\left|\eta_{ij}\right|>u\right\\}}-(\,\cdot\,)\int_{u<\left|x\right|\leq
1}x^{2}\mu(dx)\right),&\alpha\in[1,2),\end{cases}$

Case 1: $\alpha\in(0,1)$

  * Step 1.

For every $\epsilon>0$, consider the functions $s^{(u)},w^{(u)}$ and $y^{(u)}$
defined on $\tilde{\ell}_{0}$ by

(B.41) $\displaystyle s^{(u)}(\tilde{\boldsymbol{x}})$
$\displaystyle=\sum_{j}x_{j}\boldsymbol{1}_{\left\\{|x_{j}|>u\right\\}}$
(B.42) $\displaystyle w^{(u)}(\tilde{\boldsymbol{x}})$
$\displaystyle=\underset{k}{\mathsf{inf}}\sum_{j\leq
k}x_{j}\boldsymbol{1}_{\left\\{|x_{j}|>u\right\\}}$ (B.43) $\displaystyle
y^{(u)}(\tilde{\boldsymbol{x}})$
$\displaystyle=\underset{k}{\mathsf{sup}}\sum_{j\leq
k}x_{j}\boldsymbol{1}_{\left\\{|x_{j}|>u\right\\}}$

Moreover, define the mapping
$\mathcal{T}_{(u)}:\mathcal{M}_{p}\big{(}[0,1]\times\tilde{\ell}_{0}\
\left\\{\boldsymbol{0}\right\\}\big{)}\to D[0,1]$, by setting for
$\gamma=\sum_{i=1}^{\infty}\delta_{t_{i},\tilde{\boldsymbol{x}}^{i}}$,

(B.44)
$\displaystyle\mathcal{T}_{(u)}\big{(}\gamma\big{)}:=\left(\left(\sum_{T_{i}\leq
t}s^{(u)}(\tilde{\boldsymbol{x}}^{i})\right)_{T\in[0,1]},\big{\\{}T_{i}:\left\lVert\tilde{\boldsymbol{x}}^{i}\right\rVert_{\infty}>u\big{\\}},\big{\\{}I(T_{i}):\left\lVert\tilde{\boldsymbol{x}}^{i}\right\rVert_{\infty}>u\big{\\}}\right)$

where

(B.45) $\displaystyle
I(T_{i})=\sum_{T_{j}<T_{i}}s^{(u)}(\tilde{\boldsymbol{x}}^{j})+\left[\sum_{T_{k}=T_{i}}w^{(u)}(\tilde{\boldsymbol{x}}^{k}),\sum_{T_{k}=T_{i}}y^{(u)}(\tilde{\boldsymbol{x}}^{k})\right].$

Let $\Lambda=\Lambda_{1}\cap\Lambda_{2}$. Moreover, the subsets $\Lambda_{1}$
and $\Lambda_{2}$ are defined as below

$\Lambda_{1}=\\{\eta\in\mathbf{M}_{p}([0,1]\times\mathbb{E}):\eta(\\{0,1\\}\times\mathbb{E})=0=\eta([0,1]\times\\{\pm\infty,\pm
u\\})\\},\\\\[10.00002pt]
\Lambda_{2}=\\{\eta\in\mathbf{M}_{p}([0,1]\times\mathbb{E}):\eta(\\{t\\}\times(u,\infty])\cdot\eta(\\{t\\}\times[-\infty,-u))=0,\
\text{for all $t\in[0,1]$}\\}.$

  * Step 2.

Define with $W_{i}=\sum_{j\in\mathbb{Z}}|Q_{i,j}|$ and
$\sum_{i=1}^{\infty}\delta_{P_{i}W_{i}}$ is a Poisson point process on
$(0,\infty]$ with intensity measure given by
$\theta\mathbb{E}[W_{i}^{\alpha}]\alpha y^{-\alpha-1}dy$. This definition
implies that we can express the point process as below

(B.46)
$\displaystyle\sum_{i=1}^{\infty}P_{i}W_{i}=\sum_{i=1}^{\infty}\sum_{j\in\mathbb{Z}}Pi|Q_{i,j}|<\infty<\infty,\
\ \text{almost surely},$

Our aim is to prove that the above limit process is an $\alpha-$stable Lévy
process in the topological space $D([0,1],\mathbb{R})$ equipped with the
$M_{1}$ topology.

Therefore, by defining with $s(\tilde{\boldsymbol{x}})=\sum_{j}x_{j}$, we
obtain that the process can be expressed as

(B.47) $\displaystyle V(t)=\sum_{T_{i}\leq
t}s\left(P_{i}\boldsymbol{Q}_{i}\right),\ \ \ t\in[0,1]$

is almost surely a well-defined element in the topological space $D$ and thus,
is an $\alpha-$stable Lévy process. Consequently we have that the partial sum
process is equivalent to

$\displaystyle S_{n,u}(t)$ $\displaystyle=\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}\boldsymbol{1}_{\left\\{\frac{|X_{i}|}{a_{n}}>u\right\\}},\
\ \alpha\in(0,1)\ \text{and}\ t\in[0,1],$ $\displaystyle=\sum_{T_{i}\leq
t}s\left(P_{i}\boldsymbol{Q}_{i}\right)=\sum_{T_{i}\leq
t}\sum_{j\in\mathbb{Z}}P_{i}Q_{i,j}\boldsymbol{1}_{\left\\{|P_{i}Q_{i,j}|>u\right\\}},\
\ t\in[0,1].$

Notice that Step 2 above holds due to an application of the following
Corollary.

######  Corollary 2.

Let $\left\\{r_{n}\right\\}$ be a nonnegative non decreasing integer valued
sequence such that $\mathsf{lim}_{n\to\infty}r_{n}\infty$ and
$\mathsf{lim}_{n\to\infty}n/r_{n}=\infty$, and let $\left\\{a_{n}\right\\}$ be
a non decreasing sequence such that $n\mathbb{P}\big{(}|X_{0}|>a_{n}\big{)}\to
1$. Then, it holds that

(B.48) $\displaystyle
N_{n}^{\prime\prime}\sum_{i=1}^{k_{n}}\delta_{(i/k_{n},(X_{(i-1)r_{n}+1},...,X_{ir_{n}})/a_{n})}\overset{d}{\to}\sum_{i=1}^{\infty}\delta_{(T_{i},P_{i}\boldsymbol{Q}_{i})}$

where $\delta_{(T_{i},P_{i})}$ is a Poisson point process on
$[0,1]\times(0,\infty]$ with intensity measure $Leb\times d(-\theta
y^{-\alpha})$.

######  Remark 13.

Notice that the above representation holds if and only if we prove that all
the previous steps hold. In other words, we need to prove that in our settings
a similar limit result holds as in Theorem 4.5 in the paper of Basrak et al.,
(2018)

  * Step 3.

The next step, is to show that the partial sum process under consideration is
a tight process in the topological space $D([0,1],\mathbb{R})$. Notice that
since two marginal sequences of processes are tight, the joint sequences of
processes must also be tight.

  * Step 4.

The last step is to show that the original partial sum process $S_{n}$; and
therefore $V_{n}$, since $\alpha\in(0,1)$ also converges in distribution to
$V^{\prime}$, which implies that we can use the metric distance to prove that
it converges in probability to zero as $n\to\infty$.

Therefore, since the limit variable $V$ that corresponds to the limit of the
partial sum process $V_{n}$ is an $\alpha-$stable random variable, then it has
the following characteristic function representation

(B.49)
$\displaystyle\mathbf{E}\left[e^{izV(1)}\right]=\begin{cases}\displaystyle\mathsf{exp}\left\\{i\tau
z-c|z|^{\alpha}\left(1-i\beta\mathsf{sign}(z)\mathsf{tan}\left(\frac{\pi\alpha}{2}\right)\right)\right\\},&\alpha\neq
1\\\ \\\ \displaystyle\mathsf{exp}\left\\{i\tau
z-c|z|\left(1+i\beta\frac{2}{\pi}\mathsf{sign}(z)\mathsf{log}|z|\right)\right\\},&\alpha=1\end{cases}$

where $c>0$, $\beta\in[-1,1]$ and $\tau\in\mathbb{R}$, with $V(1)$ as
expression (B.38) in Theorem 3.4. By Remark 3.2 in Davis and Hsing, (1995),
the scale parameter $c$ and the symmetry parameter $\beta$ are defined with
the following coefficients respectively, (see also, Feller, (1971) pages
568-570).

(B.50) $\displaystyle{\color[rgb]{0,0,1}c}$
$\displaystyle=\begin{cases}\big{(}c_{+}+c_{-}\big{)}\frac{\displaystyle\Gamma(3-\alpha)}{\displaystyle\alpha(\alpha-1)}\mathsf{cos}\left(\frac{\pi\alpha}{2}\right),&\text{if}\
\alpha\neq 1,\\\
\big{(}c_{+}+c_{-}\big{)}\displaystyle\frac{\pi}{2},&\text{if}\
\alpha=1\end{cases}$ (B.51) $\displaystyle{\color[rgb]{0,0,1}\beta}$
$\displaystyle=\frac{c_{+}-c_{-}}{c_{+}+c_{-}},$

where

(B.52) $\displaystyle c_{+}$ $\displaystyle=\frac{\alpha}{2-\alpha}\
\underset{u\to
0}{\mathsf{lim}}\int_{0}^{\infty}\mathbb{P}\left(\sum_{j=1}^{\infty}yQ_{1j}\mathbf{1}_{\left\\{u,\infty\right\\}}(y|Q_{1j}|)>1\right)\gamma\alpha
y^{-\alpha-1}dy$ (B.53) $\displaystyle c_{+}$
$\displaystyle=\frac{\alpha}{2-\alpha}\ \underset{u\to
0}{\mathsf{lim}}\int_{0}^{\infty}\mathbb{P}\left(\sum_{j=1}^{\infty}yQ_{1j}\mathbf{1}_{\left\\{u,\infty\right\\}}(y|Q_{1j}|)<-1\right)\gamma\alpha
y^{-\alpha-1}dy$

Moreover, the location parameter $\tau$ is defined as below

(B.54) $\displaystyle\tau=\begin{cases}\underset{u\to 0}{\mathsf{lim}}\
\displaystyle\int_{-\infty}^{\infty}\frac{\displaystyle\mathsf{sin}x}{x^{2}}{\color[rgb]{0,0,1}M_{u}}(dx)-b(u,1],&\
\text{if}\ \alpha=1,\\\ \underset{u\to 0}{\mathsf{lim}}\
\displaystyle\int_{-\infty}^{\infty}x^{-1}{\color[rgb]{0,0,1}M_{u}}(dx)-b(u,1],&\
\text{if}\ \alpha>1,\end{cases}$

where $M_{u}(dx)$ is the canonical measure in Feller’s representation of an
infinitely divisible characteristic function given by

(B.55) $\displaystyle
M_{u}(dx)=x^{2}\int_{0}^{\infty}\mathbb{P}\left(\sum_{j=1}^{\infty}yQ_{1j}\mathbf{1}_{\left\\{u,\infty\right\\}}(y|Q_{1j}|)\in
dx\right)\gamma\alpha y^{-\alpha-1}dy$

Case 2: $\alpha\in[1,2)$

Furthermore, in the case when $\alpha\in[1,2)$ we need to introduce centering,
and therefore we define the cádlág process $V_{n}$ by setting $t\in[0,1]$ for

(B.56) $\displaystyle S_{n,u}(t)=\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}\mathbf{1}_{\left\\{\frac{|X_{i}|}{a_{n}}>u\right\\}}$

and the corresponding partial sum process with a centering sequence given by

(B.57) $\displaystyle V_{n,u}(t)=S_{n,u}(t)-\lfloor
nt\rfloor\mathbf{E}\left[\frac{X_{1}}{a_{n}}1_{\left\\{u<\frac{|X_{i}|}{a_{n}}\leq
1\right\\}}\right].$

######  Lemma 6.

Let $\alpha\in[1,2)$ and let the convergence of Theorem 4.5 hold. Then there
exists an $\alpha-$stable Lévy process $V$ on $[0,1]$ such that, as
$\epsilon\to 0$, the process $V_{\epsilon}$ defined below

(B.58) $\displaystyle V_{\epsilon}(t)=\sum_{T_{i}\leq
t}s^{\epsilon}(P_{i}\boldsymbol{Q}_{i})-t\int_{\left\\{x:\epsilon<|x|\leq
1\right\\}}x\mu(dx).$

converges uniformly almost surely (along some subsequence) to $V$.

###### Proof.

More precisely, we show that for all $\epsilon>0$

(B.59) $\displaystyle\int_{\left\\{x:\epsilon<|x|\leq
1\right\\}}x\mu(dx)=\theta\int_{0}^{\infty}\mathbb{E}\left[y\sum_{y\in\mathbb{Z}}Q_{j}\boldsymbol{1}_{\left\\{\epsilon<y|Q_{j}|\leq
1\right\\}}\right]\alpha y^{-\alpha-1}dy.$

Notice that it holds that

(B.60)
$\displaystyle\theta\mathbb{E}\left[\sum_{j\in\mathbb{Z}}Q_{j}\left|Q_{j}\right|^{\alpha-1}\right]=2p-1.$

Moreover, by Fubini’s theorem, if $\alpha>1$, we have that

$\displaystyle\theta\int_{0}^{\infty}\mathbb{E}\left[y\sum_{j\in\mathbb{Z}}Q_{j}\boldsymbol{1}_{\left\\{\epsilon<y|Q_{j}|\leq
1\right\\}}\right]\alpha y^{-\alpha-1}dy$
$\displaystyle=\alpha\theta\mathbb{E}\left[\sum_{j\in\mathbb{Z}}Q_{j}\int_{\epsilon|Q_{j}|^{-1}}^{|Q_{j}|^{-1}}y^{-\alpha}dy\right]$
$\displaystyle=\frac{\alpha}{\alpha-1}\left(\epsilon^{-\alpha+1}-1\right)\theta\mathbb{E}\left[\sum_{j\in\mathbb{Z}}Q_{j}|Q_{j}|^{\alpha-1}\right]$
$\displaystyle=\frac{\alpha}{\alpha-1}\left(\epsilon^{-\alpha+1}-1\right)(2p-1).$

Hence, for all $t\in[0,1]$ we have that

(B.61) $\displaystyle V_{\epsilon}(t)=\sum_{T_{i}\leq
t}s^{\epsilon}(P_{i}\boldsymbol{Q}_{i})-t\theta\int_{0}^{\infty}\mathbb{E}\left[y\sum_{j\in\mathbb{Z}}Q_{j}\boldsymbol{1}_{\left\\{\epsilon<y|Q_{j}|\leq
1\right\\}}\right]\alpha y^{-\alpha-1}dy$

Furthermore, notice that we can define with $W=\sum_{j\in\mathbb{Z}}|Q_{j}|$
and $W_{i}=\sum_{j\in\mathbb{Z}}|Q_{i,j}|$ such that $\left\\{W_{i},i\geq
1\right\\}$ is a sequence of i.i.d random variables with the same distribution
as $W$.

Therefore, for every $\delta>0$, there are almost surely at most finitely many
points $P_{i}W_{i}$ such that it holds that $P_{i}W_{i}>\delta$. For every
$\delta,\epsilon>0$, define with

(B.62) $\displaystyle
m_{\epsilon,\delta}=\theta\int_{0}^{\infty}\mathbb{E}\left[y\sum_{j\in\mathbb{Z}}Q_{j}\boldsymbol{1}_{\left\\{\epsilon<y|Q_{j}|\leq
1,\delta<yW\right\\}}\right]\alpha y^{-\alpha-1}dy$

######  Remark 14.

Notice that by the Theorem 3.1 of Basrak et al., (2018) it holds that the
finite dimensional distributions of $V_{0,\delta}$ converge to those of an
$\alpha-$stable Lévy process. Since the topological space $D([0,1])$ is
complete under its metric, we can obtain the existence of the stochastic
process $V=\left\\{V(t),t\in[0,1]\right\\}$ with continuous paths
(trajectories with independent increments), in $D([0,1])$ almost surely and
such that $\mathsf{lim}_{k\to\infty}\left\lVert
V_{0,\delta_{k}}-V\right\rVert_{\infty}=0$ almost surely.

Therefore, it only remains to prove that for all $u>0$, we have that

(B.63) $\displaystyle\underset{\delta\to 0}{\mathsf{lim}}\
\underset{\epsilon\to 0}{\mathsf{limsup}}\ \mathbb{P}\big{(}\left\lVert
V_{\epsilon}-V_{\epsilon,\delta}\right\rVert_{\infty}>u\big{)}=0.$

In particular, proving that the above result holds, then it implies that
$\left\lVert V_{\epsilon}-V_{\epsilon,\delta}\right\rVert_{\infty}\to 0$, in
probability and hence that, along some subsequence, $V_{\epsilon}$ converges
to $V$ unifornmly almost surely.

Moreover, since for $\delta\leq 1$,
$yW=\sum_{j\in\mathbb{Z}}y|Q_{j}|\leq\delta$ implies that
$y|Q_{j}|\leq\delta\leq 1$ for all $j\in\mathbb{Z}$,

(B.64) $\displaystyle V_{\epsilon}(t)-V_{\epsilon,\delta}(t)=\sum_{T_{i}\leq
t}\sum_{j\in\mathbb{Z}}PiQ_{i,j}\boldsymbol{1}_{\left\\{\epsilon<P_{i}|Q_{j}|,P_{i}W_{i}\leq\delta\right\\}}-t\theta\int_{0}^{\infty}\mathbb{E}\left[y\sum_{j\in\mathbb{Z}Q_{j}}\boldsymbol{1}_{\left\\{\epsilon<P_{i}|Q_{j}|,P_{i}W_{i}\leq\delta\right\\}}\right]$

∎

Therefore, in order to prove that indeed the coefficients that correspond to
the characteristic function of the partial sum process converge to the Lévy
process $V$ with characteristic function and corresponding coefficients that
are part of the triple, we need to employ the derivations used in Lemma 6.3 of
Davis and Hsing, (1995).

Thus, the components of the limiting process $L=\left(L_{1},L_{2}\right)$ will
be expressed as functionals of the limiting process
$N=\sum_{i}\sum_{j}\delta_{(T_{i},P_{i}\eta_{ij})}$, and the description of
the characteristic triple of $L_{1}$ and the exponent measure of $L_{2}$ will
be in terms of the measures $\nu_{1}$ and $\nu_{2}$ on $\mathbb{R}$ defined by

$\displaystyle\nu_{1}(dx)$
$\displaystyle=\big{(}c_{+}\mathbf{1}_{(0,+\infty)}(x)+c_{-}\mathbf{1}_{(-\infty,0)}(x)\big{)}\theta\alpha|x|^{-\alpha-1}dx$
$\displaystyle\nu_{2}(dx)$ $\displaystyle=r\theta\alpha
x^{-\alpha-1}\mathbf{1}_{(0,+\infty)}(x)dx$

We use the following notation

(B.65)
$\displaystyle{\color[rgb]{0,0,1}x^{\alpha^{\star}}:=x|x|^{\alpha-1}}\equiv|x|^{\alpha}\left(\mathbf{1}_{(0,\infty}(x)-\mathbf{1}_{(-\infty,0)}(x)\right)$

The following parameters were computed by Theorem 3.2 in Davis and Hsing,
(1995). Thus, we define with

(B.66) $\displaystyle
c:=\theta\mathbb{E}\left[\left|\sum_{j}\eta_{j}\right|^{\alpha}\right],\ \ \
\beta:=\frac{\mathbb{E}\left[\left(\sum_{j}\eta_{j}\right)^{\alpha^{\star}}\right]}{\mathbb{E}\left[\left|\sum_{j}\eta_{j}\right|^{\alpha}\right]}=\frac{\mathbb{E}\left[\left(\sum_{j}\eta_{j}\right)\left|\sum_{j}\eta_{j}\right|^{\alpha-1}\right]}{\mathbb{E}\left[\left|\sum_{j}\eta_{j}\right|^{\alpha}\right]}$

(B.67)
$\displaystyle\tau=\begin{cases}\displaystyle(\alpha-1)^{-1}\alpha\theta\mathbb{E}\left[\left(\sum_{j}\eta_{j}\right)^{\alpha^{\star}}\right],&\alpha>1,\\\
\displaystyle\theta\left(s\mathbb{E}\left[\sum_{j}\eta_{j}\right]-\mathbb{E}\left[\sum_{j}\eta_{j}\mathsf{log}\left(\left|\sum_{i}\eta_{i}\eta_{j}^{-1}\right|\right)\right]\right),&\alpha=1,\end{cases}$

with
$s=\int_{0}^{\infty}\left(\mathsf{sin}(x)-x\mathbf{1}_{(0,1]}(x)\right)x^{-2}dx$.
Then, the characteristic triple of the process $V$ is of the form
$(0,\sigma,b)$, where
$\sigma(dx)=\left(c_{1}\mathbf{1}_{(0,1]}(x)+c_{2}\mathbf{1}_{(-\infty,0)}(x)\right)|x|^{-(\alpha+1)}dx$
and $b=\tau-d$, with

(B.68) $\displaystyle
c_{1}=\begin{cases}\frac{\displaystyle-c(1+\beta)}{\displaystyle
2\Gamma(-\alpha)\mathsf{cos}(\pi\alpha/2)},&\alpha>1\\\ \\\
\frac{\displaystyle c(1+\beta)}{\displaystyle\pi},&\alpha=1\end{cases}\ \ \ \
\text{and}\ \ \ \
c_{2}=\begin{cases}\frac{\displaystyle-c(1-\beta)}{\displaystyle
2\Gamma(-\alpha)\mathsf{cos}(\pi\alpha/2)},&\alpha>1\\\ \\\
\frac{\displaystyle c(1-\beta)}{\displaystyle\pi},&\alpha=1\end{cases}$

Thus, for a complete proof we shall also show that $b=\gamma$ which will then
imply that the Lévy process $V$ has characteristic triple
$(0,\nu_{1},\gamma_{1})$. We leave this step for the next version of this
draft.

∎

###  Proof of Theorem 3

###### Proof.

Repeating the arguments from the proof of Theorem 1, but with the summation
functional
$\widetilde{\Phi}^{(u)}\colon\mathbf{M}_{p}([0,1]\times\mathbb{E})\to
D([0,1],\mathbb{R}^{2})$,

(B.69)
$\displaystyle\widetilde{\Phi}^{(u)}\Big{(}\sum_{i}\delta_{(t_{i},x_{i})}\Big{)}(t)=\Big{(}\sum_{t_{i}\leq
t}x_{i}\,\boldsymbol{1}_{\\{u<|x_{i}|<\infty\\}},\sum_{i}x_{i}^{2}\,\boldsymbol{1}_{\\{u<|x_{i}|<\infty\\}}\Big{)},\qquad
t\in[0,1],$

instead of $\widetilde{\Phi}^{(u)}$, we obtain, as $n\to\infty$,

(B.70) $\widetilde{L}_{n}(t):=\bigg{(}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}},\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}\bigg{)}\xrightarrow{d}\widetilde{L}:=\bigg{(}\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij},\sum_{i}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\bigg{)}$

in $D([0,1],\mathbb{R}^{2})$ with the weak $M_{1}$ topology. Similar to Lemma
3 one can show that the function $g\colon D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})\to D([0,1],\mathbb{R})$ defined by

$\displaystyle g(x,y)=\frac{x}{\sqrt{y}},y>0\ \ \ \ \text{and}\ \ \ \
g(x,y)=0,y\leq 0,$

is continuous when $D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})$ is endowed with the weak $M_{1}$ topology
and $D([0,1],\mathbb{R})$ is endowed with the standard $M_{1}$ topology. Since

(B.71) $\displaystyle\mathbb{P}\left(\widetilde{L}\in
D([0,1],\mathbb{R})\times C^{\uparrow}_{0}([0,1],\mathbb{R})\right)=1,$

an application of the continuous mapping theorem yields
$g(\widetilde{L}_{n})\xrightarrow{d}g(\widetilde{L})$ as $n\to\infty$, that is

(B.72) $\displaystyle
g\left(\widetilde{L}_{1n}(t),\widetilde{L}_{2n}(1)\right):=\frac{\widetilde{L}_{1n}(t)}{\sqrt{\widetilde{L}_{2n}(1)}}\equiv\frac{\displaystyle\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}}{\displaystyle\sqrt{\sum_{i=1}^{n}\frac{X^{2}_{i}}{a_{n}}}}\xrightarrow{d}\frac{\displaystyle\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}}{\displaystyle\sqrt{\sum_{i}\sum_{j}P_{i}^{2}\eta_{ij}^{2}}}=\frac{L_{1}(t)}{\sqrt{L_{2}(1)}},\
\ \alpha\in(0,1).$

in $D([0,1],\mathbb{R})$ endowed with the $M_{1}$ topology. ∎

###  Proof of Theorem 4

Notice that in the case of $\alpha\in[1,2)$ we still consider the joint
functional convergence of $L_{n}(t):=\big{(}L_{1n}(t),L_{2n}(t)\big{)}$.
However, a key aspect in this proof is that we employ the convergence of these
coordinates to obtain a weak limit result in $D[0,1]$ endowed with the $M_{1}$
topology for the corresponding self-normalized partial sum process. Therefore,
following similar arguments to the proof of Theorem 1, but in this case using
the summation functional
$\widetilde{\Phi}^{(u)}\colon\mathbf{M}_{p}([0,1]\times\mathbb{E})\to
D([0,1],\mathbb{R}^{2})$, we obtain that

$\displaystyle\widetilde{\Phi}^{(u)}\Big{(}\sum_{i}\delta_{(t_{i},x_{i})}\Big{)}(t)=\Big{(}\sum_{t_{i}\leq
t}x_{i}\,\boldsymbol{1}_{\\{u<|x_{i}|<\infty\\}},\sum_{i}x_{i}^{2}\,\boldsymbol{1}_{\\{u<|x_{i}|<\infty\\}}\Big{)},\qquad
t\in[0,1],$

instead of $\widetilde{\Phi}^{(u)}$, we obtain, as $n\to\infty$,

$\displaystyle\widetilde{L}_{n}(t)$
$\displaystyle:=\bigg{(}\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}-\lfloor nt\rfloor
b_{1n}^{(u)},\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}-nb_{2n}^{(u)}\bigg{)}$
(B.73) $\displaystyle\xrightarrow{d}\widetilde{L}:=\bigg{(}\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}\mathbf{1}_{\left\\{P_{i}|\eta_{ik}|>u\right\\}}-tb_{1},\sum_{i}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\mathbf{1}_{\left\\{P^{2}_{i}|\eta_{ij}^{2}|>u\right\\}}-b_{2}\bigg{)}$

in $D([0,1],\mathbb{R}^{2})$ with the weak $M_{1}$ topology. Similar to Lemma
3 one can show that the function $g\colon D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})\to D([0,1],\mathbb{R})$ defined by

$\displaystyle g(x,y)=\frac{x}{\sqrt{y}},y>0\ \ \ \ \text{and}\ \ \ \
g(x,y)=0,y\leq 0,$

is continuous when $D([0,1],\mathbb{R})\times
C^{\uparrow}_{0}([0,1],\mathbb{R})$ is endowed with the weak $M_{1}$ topology
and $D([0,1],\mathbb{R})$ is endowed with the standard $M_{1}$ topology. Since

(B.74) $\displaystyle\mathbb{P}\left(\widetilde{L}\in
D([0,1],\mathbb{R})\times C^{\uparrow}_{0}([0,1],\mathbb{R})\right)=1,$

an application of the continuous mapping theorem yields
$g(\widetilde{L}_{n})\xrightarrow{d}g(\widetilde{L})$ as $n\to\infty$, that is

$\displaystyle
g\left(\widetilde{L}_{1n}(t),\widetilde{L}_{2n}(1)\right):=\frac{\widetilde{L}_{1n}(t)}{\sqrt{\widetilde{L}_{2n}(1)}}$
$\displaystyle\equiv\frac{\displaystyle\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}-\lfloor nt\rfloor
b_{1n}^{(u)}}{\displaystyle\sqrt{\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}-nb_{2n}^{(u)}}}=\frac{\displaystyle\sum_{i=1}^{\lfloor
nt\rfloor}\frac{X_{i}}{a_{n}}-\lfloor
nt\rfloor\mathbb{E}\left[\frac{X_{1}}{a_{n}}\mathbf{1}_{\left\\{{\color[rgb]{1,0,0}u<}\frac{|X_{1}|}{a_{n}}\leq
1\right\\}}\right]}{\displaystyle\sqrt{\sum_{i=1}^{n}\frac{X_{i}^{2}}{a_{n}^{2}}-n\mathbb{E}\left[\frac{X^{2}_{1}}{a^{2}_{n}}\mathbf{1}_{\left\\{u<\frac{X_{1}^{2}}{a_{n}}\leq
1\right\\}}\right]}},\ \ \alpha\in[1,2)$
$\displaystyle\xrightarrow{d}\frac{\displaystyle\underset{u\to
0}{\mathsf{lim}}\left\\{\sum_{T_{i}\leq
t}\sum_{j}P_{i}\eta_{ij}\mathbf{1}_{\left\\{P_{i}|\eta_{ik}|>u\right\\}}-t\int_{u<|x|\leq
1}x\mu(dx)\right\\}}{\displaystyle\sqrt{\sum_{T_{i}\leq
1}\sum_{j}P_{i}^{2}\eta_{ij}^{2}\mathbf{1}_{\left\\{P^{2}_{i}|\eta_{ij}^{2}|>u\right\\}}-\int_{u<x^{2}\leq
1}x^{2}\mu(dx)}}=\frac{\widetilde{L}_{1}(\,t\,)}{\sqrt{\widetilde{L}_{2}(1)}},\
\text{as}\ n\to\infty.$

in $D([0,1],\mathbb{R})$ endowed with the $M_{1}$ topology.

The last equality follows by the joint weak convergence of the functionals
$\widetilde{L}_{1n}(\,\cdot\,)\xrightarrow{d}\widetilde{L}_{1}(\,\cdot\,)$ and
$\widetilde{L}_{2n}(\,\cdot\,)\xrightarrow{d}\widetilde{L}_{2}(\,\cdot\,)$ and
the fact that $g(\widetilde{L}_{n})\xrightarrow{d}g(\widetilde{L})$ as
$n\to\infty$. In particular, note that the limit both on the numerator as well
as in the denominator that correspond to the case in which $\alpha\in[1,2)$
hold almost surely uniformly in $[0,1]$. Moreover, in the case of $\alpha=1$,
additionally we assume that Conditions (3) and (4). These two conditions
although challenging to prove in the case of dependent data, ensure that our
main results hold for more general dependence structures as in the case of
point processes. A special type of dependence one can consider is the case of
$\rho-$mixing which those two conditions can be easily established.

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