# Estimation of cluster functionals for regularly varying time series: runs
estimators

Youssouph Cissokho University of Ottawa Rafał Kulik University of Ottawa

###### Abstract

Cluster indices describe extremal behaviour of stationary time series. We
consider runs estimators of cluster indices. Using a modern theory of
multivariate, regularly varying time series, we obtain central limit theorems
under conditions that can be easily verified for a large class of models. In
particular, we show that blocks and runs estimators have the same limiting
variance.

##  1 Introduction

Consider a stationary, regularly varying $\mathbb{R}^{d}$-valued time series
$\boldsymbol{X}=\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$. We are interested in
its extremal behaviour. A classical approach to this problem is to calculate
the extremal index. If $|\cdot|$ is an arbitrary norm on $\mathbb{R}^{d}$,
then the extremal index $\theta$ (if exists) of
$\\{|\boldsymbol{X}_{j}|,j\in\mathbb{Z}\\}$ is defined as a parameter in the
limiting distribution of the maxima. With $Q$ being the quantile function of
$|\boldsymbol{X}_{0}|$ and $a_{n}=Q(1-1/n)$ we have

$\displaystyle\lim_{n\to\infty}\mathbb{P}(a_{n}^{-1}\max_{j=1,\ldots,n}\\{|\boldsymbol{X}_{1}|,\ldots,|\boldsymbol{X}_{n}|\\}\leq
x)=\exp(-\theta x^{-\alpha})\;,\ \ x>0\;.$

The parameter $\theta\in(0,1]$ indicates the amount of clustering, with
$\theta=1$ (the case of extremal independence) meaning no-clustering of large
values.

The extremal index is just one parameter that describes clustering of
extremes. Informally speaking, it arises as the limit

$\lim_{n\to\infty}\frac{\mathbb{E}[H((\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r_{n}})/u_{n})]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;,$

for the particular choice of function
$H:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$, and a suitable choice of the
scaling sequence $u_{n}\to\infty$ and the block size $r_{n}\to\infty$.
(Formally speaking, $(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r_{n}})$ is a
random element of $(\mathbb{R}^{d})^{r_{n}}$, while the domain of $H$ is
$(\mathbb{R}^{d})^{\mathbb{Z}}$. This inconsistency will be explained later).

In particular, the extremal index is achieved by applying a suitable
functional to a cluster:

$\displaystyle
H((\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r_{n}})/u_{n})=\mathbbm{1}{\left\\{\max\\{\left|\boldsymbol{X}_{1}\right|,\ldots,\left|\boldsymbol{X}_{r_{n}}\right|\\}>u_{n}\right\\}}\;.$

That is,

$\displaystyle\theta=\lim_{n\to\infty}\frac{\mathbb{P}(\max\\{\left|\boldsymbol{X}_{1}\right|,\ldots,\left|\boldsymbol{X}_{r_{n}}\right|\\}>u_{n})}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;.$
(1.1)

Informally speaking, a cluster is a triangular array
$(\boldsymbol{X}_{1}/u_{n},\ldots,\boldsymbol{X}_{r_{n}}/u_{n})$ with
$r_{n},u_{n}\to\infty$ that converges in distribution in a certain sense.
Cluster indices are obtained by applying the appropriate functional $H$ to the
cluster. The functionals are defined on $(\mathbb{R}^{d})^{\mathbb{Z}}$, the
space of $\mathbb{R}^{d}$-valued sequences, and are such that their values do
not depend on coordinates that are equal to zero. More precisely, for
$\boldsymbol{X}=\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$
and $i\leq j\in\mathbb{Z}$, we denote
$\boldsymbol{X}_{i,j}=(\boldsymbol{X}_{i},\ldots,\boldsymbol{X}_{j})\in(\mathbb{R}^{d})^{(j-i+1)}$.
Then, we identify $H(\boldsymbol{X}_{i,j})$ with
$H(({\boldsymbol{0}},\boldsymbol{X}_{i,j},{\boldsymbol{0}}))$, where
${\boldsymbol{0}}\in(\mathbb{R}^{d})^{\mathbb{Z}}$ is the zero sequence. Such
functionals $H$ will be called cluster functionals.

Let $\left|\cdot\right|$ be an arbitrary norm on $\mathbb{R}^{d}$ and
$\\{u_{n}\\}$, $\\{r_{n}\\}$ be such that

$\displaystyle\lim_{n\to\infty}u_{n}=\lim_{n\to\infty}r_{n}=\lim_{n\to\infty}n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})=\infty\;,$
$\displaystyle\lim_{n\to\infty}r_{n}/n=\lim_{n\to\infty}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})=0\;.$
($\mathcal{R}(r_{n},u_{n})$)

Given a cluster functional $H$ on $(\mathbb{R}^{d})^{\mathbb{Z}}$, we want to
estimate the limiting quantity

$\displaystyle{\boldsymbol{\nu}}^{*}(H)=\lim_{n\to\infty}{\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)=\lim_{n\to\infty}\frac{\mathbb{E}[H(\boldsymbol{X}_{1,r_{n}}/u_{n})]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;.$
(1.2)

To guarantee existence of the limit we will require additional anticlustering
assumptions on the time series $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$. For
$\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$
define $\boldsymbol{x}^{*}=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|$. The
cluster indices of interest are, among others:

  * •

the extremal index obtained with
$H_{1}(\boldsymbol{x})=\mathbbm{1}{\left\\{\boldsymbol{x}^{*}>1\right\\}}$,
$\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$;

  * •

the cluster size distribution obtained with

$\displaystyle
H_{2}(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}=m\right\\}}\;,\
\
\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}\;,\
\ m\in\mathbb{N}\;;$ (1.3)

  * •

the stop-loss index of a univariate time series obtained with

$\displaystyle
H_{3}(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j\in\mathbb{Z}}(x_{j}-1)_{+}>\eta\right\\}}\;,\
\
\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in\mathbb{R}^{\mathbb{Z}}\;,\
\ \eta>0\;;$ (1.4)

  * •

the large deviation index of a univariate time series obtained with

$\displaystyle
H_{4}(\boldsymbol{x})=\mathbbm{1}{\left\\{K(\boldsymbol{x})>1\right\\}}\;,\ \
K(\boldsymbol{x})=\left(\sum_{j\in\mathbb{Z}}x_{j}\right)_{+}\;,\ \
\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in\mathbb{R}^{\mathbb{Z}}\;;$
(1.5)

  * •

the ruin index of a univariate time series obtained with

$\displaystyle
H_{5}(\boldsymbol{x})=\mathbbm{1}{\left\\{K(\boldsymbol{x})>1\right\\}}\;,\ \
K(\boldsymbol{x})=\sup_{i\in\mathbb{Z}}\left(\sum_{j\leq
i}x_{j}\right)_{+}\;,\ \
\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in\mathbb{R}^{\mathbb{Z}}\;.$
(1.6)

As indicated above, the extremal index is the classical quantity that arises
in the extreme value theory for dependent sequences. Similarly, the cluster
size distribution has been studied in [Hsi91] and [DR10]. The large deviation
index was studied under the name cluster index in [MW13, MW14]. It quantifies
the effect of dependence in large deviations results.

Several methods of estimation of the limit ${\boldsymbol{\nu}}^{*}(H)$ in
(1.2) may be employed. The natural one is to consider a statistics based on
disjoint blocks of size $r_{n}$, cf. [DR10] and [KS20],

$\displaystyle{\widetilde{\boldsymbol{\nu}}}^{*}_{n,r_{n}}(H):=\frac{1}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{m_{n}}H(\boldsymbol{X}_{(i-1)r_{n}+1,ir_{n}}/u_{n})\;,$

where $m_{n}=[n/r_{n}]$ is the number of disjoint blocks. The data-based
estimator is constructed as follows. Let $k_{n}\to\infty$ be a sequence of
integers and define $u_{n}$ by
$k_{n}=n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$. Let
${\left|\boldsymbol{X}\right|}_{(n:1)}\leq\cdots\leq{\left|\boldsymbol{X}\right|}_{(n:n)}$
be order statistics from
$\left|\boldsymbol{X}_{1}\right|,\ldots,\left|\boldsymbol{X}_{n}\right|$.
Define

$\displaystyle\widehat{\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)$
$\displaystyle:=\frac{1}{k_{n}}\sum_{i=1}^{m_{n}}H(\boldsymbol{X}_{(i-1)r_{n}+1,ir_{n}}/{\left|\boldsymbol{X}\right|}_{(n:n-k_{n})})\;.$
(1.7)

The general asymptotic theory for disjoint blocks estimators was developed in
[DR10]. See also [KS20, Chapter 10]. The limiting variance of the disjoint
blocks estimator can be represented as

$\displaystyle{\boldsymbol{\nu}}^{*}$
$\displaystyle(\\{H-{\boldsymbol{\nu}}^{*}(H)\mathcal{E}\\}^{2})\;,$ (1.8)

where
$\mathcal{E}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}$.
This result was established (implicitly) in [DR10], but the form of the
limiting variance is again given in [KS20, Chapter 10].

Another approach to estimation of ${\boldsymbol{\nu}}^{*}(H)$ is to consider
the sliding blocks statistics

$\displaystyle{\widetilde{\boldsymbol{\mu}}}^{*}_{n,r_{n}}(H):=\frac{1}{q_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=0}^{q_{n}-1}H\left(\boldsymbol{X}_{i+1,i+r_{n}}/u_{n}\right)\;$
(1.9)

and and the corresponding estimator defined in terms of order statistics:

$\displaystyle\widehat{\boldsymbol{\mu}}^{*}_{n,r_{n}}(H)=\frac{1}{r_{n}k_{n}}\sum_{i=0}^{q_{n}-1}H\left(\boldsymbol{X}_{i+1,i+r_{n}}/{\left|\boldsymbol{X}\right|}_{(n:n-k_{n})}\right)\;.$
(1.10)

Here, $q_{n}=n-r_{n}-1$ is the number of sliding blocks. In [DN20] the authors
used the framework of [DR10] and showed that the limiting variance of the
sliding blocks estimator never exceeds that of the disjoint blocks estimator.
In case of the extremal index, both variances were proven to be equal. In
[CK21] it was shown that the limiting variances for both disjoint and sliding
blocks estimators agree and are given by the expression in (1.8) for an
arbitrary choice of $H$. We note at this point that the methodology used in
[DR10, DN20, KS20, CK21] fits into Peak Over Threshold (PoT) framework. On the
other hand, in the Block Maxima (BM) framework, sliding blocks estimators
yield typically smaller variance; see [BS18b, BS18a]. As of this moment, there
is no thorough explanation of these phenomena and no formal comparison between
PoT and BM framework. See [FdH15] for some partial results and [BZ18] for a
recent review.

In the present paper we are interested in the so-called runs estimators. In
the context of the extremal index, this approach goes back to [WN98] and stems
from the following representation of the extremal index:

$\displaystyle\theta=\lim_{n\to\infty}\mathbb{P}(\max\\{\left|\boldsymbol{X}_{1}\right|,\ldots,\left|\boldsymbol{X}_{r_{n}}\right|\\}\leq
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\;.$ (1.11)

We note that

$\displaystyle\theta=\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}[\mathbbm{1}{\left\\{\mathcal{A}(({\boldsymbol{X}_{0}},\ldots,\boldsymbol{X}_{r_{n}})/u_{n})=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{0}\right|>u_{n}\right\\}}]\;,$

where

$\displaystyle\mathcal{A}(\boldsymbol{x})=\sup\\{j:\left|\boldsymbol{x}_{j}\right|>1\\}\;$

gives the position of the last exceedence above 1 in a particular block.
Recall again the convention
$\mathcal{A}(({\boldsymbol{X}_{0}},\ldots,\boldsymbol{X}_{r_{n}})/u_{n})=\mathcal{A}(({\boldsymbol{0}},{\boldsymbol{X}_{0}},\ldots,\boldsymbol{X}_{r_{n}})/u_{n})$.
Then, $\mathcal{A}$ is an example of so-called anchoring map. Special cases of
anchoring maps were considered in [Has18] and [BP18], while in [KS20] their
connection to cluster indices ${\boldsymbol{\nu}}^{*}(H)$ was thoroughly
investigated. It turns out that with an arbitrary choice of the anchoring map
$\mathcal{A}$ we have

$\displaystyle{\boldsymbol{\nu}}^{*}(H)=\mathbb{E}[H^{\mathcal{A}}(\boldsymbol{Y})]\;,$

where

$\displaystyle
H^{\mathcal{A}}(\boldsymbol{x})=H(\boldsymbol{x})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x})=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>1\right\\}}\;.$

This motivates the following runs statistics:

$\displaystyle{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})=\frac{1}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=r_{n}+1}^{n-r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;.$
(1.12)

Indeed, under the appropriate conditions, Proposition 2.7 gives

$\displaystyle\lim_{n\to\infty}\mathbb{E}[{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})]=\mathbb{E}[H^{\mathcal{A}}(\boldsymbol{Y})]={\boldsymbol{\nu}}^{*}(H)\;.$

The data-based runs estimator is then

$\displaystyle\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})=\frac{1}{k_{n}}\sum_{i=r_{n}+1}^{n-r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/{\left|\boldsymbol{X}\right|}_{(n:n-k)}\right)\;.$

The main result of this paper is Theorem 3.5, the asymptotic normality of the
appropriately normalized estimator
$\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$. We show, in
particular, that the limiting variance agrees with the one for the disjoint
blocks and sliding blocks estimators; cf. [DR10], [KS20, Chapter 10], [CK21].
Furthermore, we prove that we cannot achieve variance reduction by considering
a linear combination of runs estimators with a different choice of anchoring
maps $\mathcal{A}$ and $\tilde{\mathcal{A}}$. Indeed, it turns out that
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$ and
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\tilde{\mathcal{A}}})$ are
totally dependent in the limit. We note in passing that even though general
ideas of proofs are similar to those of [CK21], however, technicalities are
significantly different. Differences stem primarily from conditioning on
$\\{\left|\boldsymbol{X}_{j}\right|>u_{n}\\}$ used in case of the runs
estimators.

Thus, from the theoretical point of view the limiting behaviour of all
(disjoint blocks, sliding blocks, runs) estimators is the same. However, for
finite samples a bias has to be taken into account. We note first that the
theoretical finite-sample bias for both disjoint and sliding blocks estimators
is the same. This can be also seen in extensive simulation studies in [CK21].
On the other hand, we were not able to get an useful formula for the bias in
the runs estimator case. As such we relied on simulations. It turns out that
runs estimator are typically heavily biased when estimation of the extremal
index is concerned. However, the runs estimators may have an advantage when
other cluster indices are considered.

The paper is structured as follows. Section 2 contains definitions, notation
and preliminary results on convergence of clusters. It is primarily based on
[KS20, Chapters 5 and 6], with some results from [BS09], [BPS18], [PS18].
Section 3 defines runs pseudo-estimators and estimators. The main result of
the paper is the central limit theorem for runs estimators in Theorem 3.5. We
note again that the limiting variance agrees with the one for disjoint and
sliding blocks estimators. Simulations are performed in Section 4, while all
the proofs are contained in Section 5.

##  2 Preliminaries

In this section we fix the notation and introduce the relevant classes of
functions. In Section 2.3 we recall the notion of the tail and the spectral
tail process (cf. [BS09]). Section 2.4 introduces anchoring maps (cf. [BP18],
[Has18]). In Section 2.5 we define cluster indices. We refer to [KS20, Chapter
5] for more details. In Section 2.6 we discuss convergence of the cluster
measure, following [KS20, Chapter 6].

The most important conclusion of these preliminaries is a representation of
the cluster index ${\boldsymbol{\nu}}^{*}(H)$ (cf. (1.2)) as
$\mathbb{E}[H^{\mathcal{A}}(\boldsymbol{Y})]$, with $H^{\mathcal{A}}$ defined
in (2.6) and $\boldsymbol{Y}$ being the tail process. Also, Proposition 2.7 on
conditional weak convergence and Propositions 2.9 and 2.11 on unconditional
weak convergence play a central role in the rest of the paper.

###  2.1 Notation

Let $|\cdot|$ be a norm on $\mathbb{R}^{d}$. For a sequence
$\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$
and $i\leq j\in\mathbb{Z}\cup\\{-\infty,\infty\\}$ we denote
$\boldsymbol{x}_{i,j}=(\boldsymbol{x}_{i},\ldots,\boldsymbol{x}_{j})\in(\mathbb{R}^{d})^{j-i+1}$,
$\boldsymbol{x}_{i,j}^{*}=\max_{i\leq l\leq j}|\boldsymbol{x}_{l}|$ and
$\boldsymbol{x}^{*}=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|$. By
${\boldsymbol{0}}$ we denote the zero sequence; its dimension can be different
in each of its occurrences.

By $\ell_{0}(\mathbb{R}^{d})$ we denote the set of $\mathbb{R}^{d}$-valued
sequences which tend to zero at infinity. Likewise, $\ell_{1}(\mathbb{R}^{d})$
consists of sequences such that
$\sum_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|<\infty$.

###  2.2 Classes of functions

Functionals $H$ are defined on $\ell_{0}(\mathbb{R}^{d})$ with the convention
$H(\boldsymbol{x}_{i,j})=H(({\boldsymbol{0}},\boldsymbol{x}_{i,j},{\boldsymbol{0}}))$.
For $s>0$, the function $H_{s}:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$ is
defined by $H_{s}(\boldsymbol{x})=H(\boldsymbol{x}/s)$. We consider the
following classes:

  * •

$\mathcal{L}$ is the class of bounded real-valued functions defined on
$(\mathbb{R}^{d})^{\mathbb{Z}}$ that are either Lipschitz continuous with
respect to the uniform norm or almost surely continuous with respect to the
distribution of the tail process $\boldsymbol{Y}$. This class includes
functions like $\mathbbm{1}{\left\\{\boldsymbol{x}^{*}>1\right\\}}$,
$\mathbbm{1}{\left\\{\sum_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|>1\right\\}}$.
See Remark 6.1.6 in [KS20].

  * •

$\mathcal{A}\subset\mathcal{L}$ is the class of shift-invariant functionals
with support separated from ${\boldsymbol{0}}$. In particular, for
$H\in\mathcal{A}$, $H({\boldsymbol{0}})=0$. The class $\mathcal{A}$ includes
$\mathbbm{1}{\left\\{\boldsymbol{x}^{*}>1\right\\}}$.

  * •

$\mathcal{K}$ is the class of shift-invariant functionals
$K:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$ defined on
$\ell_{1}(\mathbb{R}^{d})$ such that $K({\boldsymbol{0}})=0$ and which are
Lipschitz continuous with constant $L_{K}$, i.e.

$\displaystyle|K(\boldsymbol{x})-K(\boldsymbol{y})|\leq
L_{K}\sum_{j\in\mathbb{Z}}\left|\boldsymbol{x}_{j}-\boldsymbol{y}_{j}\right|\;,\
\ \boldsymbol{x},\boldsymbol{y}\in\ell_{1}(\mathbb{R}^{d})\;.$

  * •

$\mathcal{B}\subset\mathcal{L}$ is the class of functionals $H$ of the form
$H=\mathbbm{1}{\left\\{K>1\right\\}}$, where $K\in\mathcal{K}$. Functionals in
$\mathcal{B}$ may have support which is not separated from ${\boldsymbol{0}}$.
The typical example is
$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j}|\boldsymbol{x}_{j}|>1\right\\}}$;
note that $H\not\in\mathcal{A}$.

We will also need the map $\mathcal{E}$ is defined on
$\ell_{0}(\mathbb{R}^{d})$ by
$\mathcal{E}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}$.
Note that $\mathcal{E}$ is shift-invariant, with the support separated from
zero, but is not bounded.

###  2.3 Tail and spectral tail process

Let $\boldsymbol{X}=\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ be a stationary,
regularly varying time series with values in $\mathbb{R}^{d}$ and tail index
$\alpha$. In particular,

$\displaystyle\lim_{x\to\infty}\frac{\mathbb{P}(|\boldsymbol{X}_{0}|>tx)}{\mathbb{P}(|\boldsymbol{X}_{0}|>x)}=t^{-\alpha}$

for all $t>0$. Then, there exists a sequence
$\boldsymbol{Y}=\\{\boldsymbol{Y}_{j},j\in\mathbb{Z}\\}$ such that

$\displaystyle\mathbb{P}(x^{-1}(\boldsymbol{X}_{i},\dots,\boldsymbol{X}_{j})\in\cdot\mid|\boldsymbol{X}_{0}|>x)\mbox{
converges weakly to
}\mathbb{P}((\boldsymbol{Y}_{i},\dots,\boldsymbol{Y}_{j})\in\cdot)$

as $x\to\infty$ for all $i\leq j\in\mathbb{Z}$. We call $\boldsymbol{Y}$ the
tail process. See [BS09]. We note that, in particular, $|\boldsymbol{Y}_{0}|$
has Pareto distribution with the density $\alpha x^{-\alpha-1}$, $x>1$. As
such, it follows automatically that
$\boldsymbol{Y}^{*}=\sup_{j\in\mathbb{Z}}|\boldsymbol{Y}_{j}|>1$.
Equivalently, viewing $\boldsymbol{X}$ and $\boldsymbol{Y}$ as random elements
with values in $(\mathbb{R}^{d})^{\mathbb{Z}}$, we have for every bounded or
non-negative functional $H$ on $(\mathbb{R}^{d})^{\mathbb{Z}}$, continuous
with respect to the product topology,

$\displaystyle\lim_{x\to\infty}\frac{\mathbb{E}[H(x^{-1}\boldsymbol{X})\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{0}\right|>x\right\\}}]}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>x)}$
$\displaystyle=\mathbb{E}[H(\boldsymbol{Y})]\;.$

The spectral tail process $\\{\boldsymbol{\Theta}_{j},j\in\mathbb{Z}\\}$ is
defined by
$\boldsymbol{\Theta}=\left|\boldsymbol{Y}_{0}\right|^{-1}\boldsymbol{Y}$ and
is independent of the tail process $\boldsymbol{Y}$.

###  2.4 Anchoring maps

######  Definition 2.1 (Anchoring map).

A measurable map
$\mathcal{A}:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{Z}\cup\\{-\infty,\infty\\}$
is called an anchoring map if the following two properties hold:

  * An(i):

$\mathcal{A}(\boldsymbol{x})=j$ implies
$\left|\boldsymbol{x}_{j}\right|\geq\left|\boldsymbol{x}_{0}\right|\wedge 1$;

  * An(ii):

$\mathcal{A}(B\boldsymbol{x})=\mathcal{A}(\boldsymbol{x})+1$, where $B$ is a
backsift operator.

Three basic examples of anchoring maps are:

  * •

The infargmax functional:
$\mathcal{A}^{(0)}(\boldsymbol{y})=\inf\\{j:\boldsymbol{y}_{-\infty,{j}}^{*}=\boldsymbol{y}^{*}\\}$;

  * •

The first exceedence above one:
$\mathcal{A}^{(1)}(\boldsymbol{x})=\inf\\{j:\left|\boldsymbol{x}_{j}\right|>1\\}$;

  * •

The last exceedence above one:
$\mathcal{A}^{(2)}(\boldsymbol{x})=\sup\\{j:\left|\boldsymbol{x}_{j}\right|>1\\}$.

In what follows we use the convention $\inf\emptyset=+\infty$. We note that
$\mathcal{A}^{(0)}$ is $0$-homogeneous, while
$\mathcal{A}_{s}^{(1)}(\boldsymbol{x})=\mathcal{A}^{(1)}(\boldsymbol{x}/s)$
and
$\mathcal{A}_{s}^{(2)}(\boldsymbol{x})=\mathcal{A}^{(2)}(\boldsymbol{x}/s)$
are increasing and decreasing in $s$, respectively, but they are not
$0$-homogeneous. This will play a role in the proofs.

A special importance is given to the time index 0. In particular,

  * •

If $\mathcal{A}^{(0)}(\boldsymbol{x})=0$, then
$\boldsymbol{x}_{-\infty,-1}^{*}<\left|\boldsymbol{x}_{0}\right|$ and
$\boldsymbol{x}_{1,\infty}^{*}\leq\left|\boldsymbol{x}_{0}\right|$;

  * •

If $\mathcal{A}^{(1)}(\boldsymbol{x})=0$, then
$\boldsymbol{x}^{*}_{-\infty,-1}\leq 1$ and
$\left|\boldsymbol{x}_{0}\right|>1$;

  * •

If $\mathcal{A}^{(2)}(\boldsymbol{x})=0$, then
$\boldsymbol{x}^{*}_{1,\infty}\leq 1$ and $\left|\boldsymbol{x}_{0}\right|>1$.

Applying an anchoring map to a finite block, say $\boldsymbol{x}_{-r,r}$ with
$r\in\mathbb{N}$, is equivalent to applying it
$\boldsymbol{x}=({\boldsymbol{0}},\boldsymbol{x}_{-r,r},{\boldsymbol{0}})$.
For example, $\mathcal{A}^{(0)}(\boldsymbol{x}_{-r,r})=0$ means that
$\boldsymbol{x}_{-r,-1}^{*}<\left|\boldsymbol{x}_{0}\right|$ and
$\boldsymbol{x}_{1,r}^{*}\leq\left|\boldsymbol{x}_{0}\right|$. This in turn
implies also that $\mathcal{A}^{(0)}(\boldsymbol{x}_{-s,s})=0$ for $0<s<r$.
Similarly,

$\displaystyle\mathcal{A}(\boldsymbol{x}_{-r,r})=0\Rightarrow\mathcal{A}(\boldsymbol{x}_{-s,s})=0\;,\
\ 0<s<r$

for $\mathcal{A}=\mathcal{A}^{(1)},\mathcal{A}^{(2)}$. However, we do not know
if this important property (used explicitly in the proofs) holds for any
anchoring map. As such, in the paper we focus on the three anchoring maps
introduced above.

Furthermore, note that An(ii) gives that

$\displaystyle\mathcal{A}(\boldsymbol{x}_{h-r,h+r})=h\Leftrightarrow\mathcal{A}(B^{-h}\boldsymbol{x}_{-r,r})=0\;.$
(2.1)

Indeed, consider for example $\mathcal{A}^{(0)}$. Then
$\mathcal{A}^{(0)}(\boldsymbol{x}_{h-r,h+r})=h$ means that
$\boldsymbol{x}^{*}_{h-r,{h}-1}<\left|\boldsymbol{x}_{h}\right|$ and
$\boldsymbol{x}_{h+1,h+r}\leq\left|\boldsymbol{x}_{h}\right|$. Set
$\tilde{\boldsymbol{x}}=B^{-h}\boldsymbol{x}$, so that
$\tilde{\boldsymbol{x}}_{-r}=\boldsymbol{x}_{h-r}$,
$\tilde{\boldsymbol{x}}_{0}=\boldsymbol{x}_{h}$ and
$\tilde{\boldsymbol{x}}_{r}=\boldsymbol{x}_{h+r}$. Thus,
$\tilde{\boldsymbol{x}}_{-r,-1}<\left|\tilde{\boldsymbol{x}}_{0}\right|$ and
$\tilde{\boldsymbol{x}}_{1,r}\leq\left|\tilde{\boldsymbol{x}}_{0}\right|$.
This in turn means that
$\mathcal{A}^{(0)}(\widetilde{\boldsymbol{x}}_{-r,r})=\mathcal{A}^{(0)}(B^{-h}\boldsymbol{x}_{-r,r})=0$.

###  2.5 Cluster measure and cluster indices

Let $\mathcal{A}$ be an anchoring map. If
$\mathbb{P}(\mathcal{A}(\boldsymbol{Y})\notin\mathbb{Z})=0$ then we can define

$\displaystyle{\vartheta}=\mathbb{P}(\mathcal{A}(\boldsymbol{Y})=0)\;.$ (2.2)

We want to emphasize that ${\vartheta}$ does not depend on the choice of the
anchoring map (see [PS18] and [KS20, Theorem 5.4.2]). In particular,

$\displaystyle{\vartheta}=\mathbb{P}(\mathcal{A}^{(1)}(\boldsymbol{Y})=0)=\mathbb{P}\left(\sup_{j\leq-1}|\boldsymbol{Y}_{j}|\leq
1\right)=\mathbb{P}(\mathcal{A}^{(2)}(\boldsymbol{Y})=0)=\mathbb{P}\left(\sup_{j\geq
1}|\boldsymbol{Y}_{j}|\leq 1\right)\;.$

The above identity follows from the time-change formula, see [CK21, Section
7.1]. Therefore, ${\vartheta}$ can be recognized as the (candidate) extremal
index. It becomes the usual extremal index under additional mixing and
anticlustering conditions (cf. Section 7.5 in [KS20]).

Recall that
$\mathcal{E}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}$.
The property An(i) of the anchoring maps implies

$\displaystyle\sum_{h\in\mathbb{Z}}\mathbb{P}(\mathcal{A}(\boldsymbol{Y})=h)\leq\sum_{h\in\mathbb{Z}}\mathbb{P}(\left|\boldsymbol{Y}_{h}\right|>1)\;.$
(2.3)

By [KS20, Lemma 9.2.3] the latter series is finite if an appropriate
anticlustering condition holds (see $\mathcal{S}(r_{n},u_{n})$ to be
introduced later on).

######  Definition 2.2 (Cluster measure).

Let $\boldsymbol{Y}$ and $\boldsymbol{\Theta}$ be the tail process and the
spectral tail process, respectively, such that
$\mathbb{P}(\lim_{|j|\to\infty}\boldsymbol{Y}_{j}={\boldsymbol{0}})=1$. The
cluster measure is the measure ${\boldsymbol{\nu}}^{*}$ on
$\ell_{0}(\mathbb{R}^{d})$ defined by

$\displaystyle{\boldsymbol{\nu}}^{*}={\vartheta}\int_{0}^{\infty}\mathbb{E}[\delta_{r\boldsymbol{\Theta}}\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{\Theta})=0\right\\}}]\alpha
r^{-\alpha-1}\mathrm{d}r\;.$ (2.4)

The measure ${\boldsymbol{\nu}}^{*}$ is boundedly finite on
$(\mathbb{R}^{d})^{\mathbb{Z}}\setminus\\{{\boldsymbol{0}}\\}$, puts no mass
at ${\boldsymbol{0}}$ and is $\alpha$-homogeneous.

Furthermore, the cluster measure can be expressed in terms of another
sequence.

######  Definition 2.3.

Assume that $\mathbb{P}(\mathcal{A}(\boldsymbol{Y})\notin\mathbb{Z})=0$. The
conditional spectral tail process $\boldsymbol{Q}$ is a random sequence with
the distribution of $(\boldsymbol{Y}^{*})^{-1}\boldsymbol{Y}$ conditionally on
$\mathcal{A}(\boldsymbol{Y})=0$.

The sequence $\boldsymbol{Q}$ appeared implicitly in the seminal paper [DH95].
See also [BS09], [PS18, Definition 3.5] and [KS20, Chapter 5]. An abstract
setting is considered in [DHS18].

Note for example that $\mathcal{A}^{(0)}(\boldsymbol{Y})=0$ gives
$\boldsymbol{Y}^{*}=\left|\boldsymbol{Y}_{0}\right|$. Thus, (2.4) and the
definition of $\boldsymbol{Q}$ give for a bounded or non-negative measurable
function $H$ on $\ell_{0}(\mathbb{R}^{d})$ (see Definition 5.4.11 in [KS20]),

$\displaystyle{\boldsymbol{\nu}}^{*}(H)$
$\displaystyle=\vartheta\int_{0}^{\infty}\mathbb{E}[H(r\boldsymbol{Q})]\alpha
r^{-\alpha-1}\mathrm{d}r\;.$

If moreover $H$ is such that $H(\boldsymbol{y})=0$ if
$\boldsymbol{y}^{*}\leq\epsilon$ for one $\epsilon>0$, then

$\displaystyle{\boldsymbol{\nu}}^{*}(H)=\epsilon^{-\alpha}\mathbb{E}[H(\epsilon\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}]\;.$
(2.5)

For a shift-invariant $H:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$ and an
anchoring map $\mathcal{A}$ define

$\displaystyle
H^{\mathcal{A}}(\boldsymbol{x})=H(\boldsymbol{x})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x})=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>1\right\\}}\;.$
(2.6)

Thus, since $\left|\boldsymbol{Y}_{0}\right|>1$, if $H$ is such that
$H(\boldsymbol{y})=0$ whenever $\boldsymbol{y}^{*}\leq 1$, then (2.5) gives

$\displaystyle{\boldsymbol{\nu}}^{*}(H)=\mathbb{E}[H^{\mathcal{A}}(\boldsymbol{Y})]\;.$
(2.7)

Note that the ${\boldsymbol{\nu}}^{*}(H)$ does not agree with
$\mathbb{E}[H(\boldsymbol{Y})]$.

######  Definition 2.4 (Cluster index).

We will call ${\boldsymbol{\nu}}^{*}(H)$ the cluster index associated to the
functional $H$.

###  2.6 Convergence of cluster measure

Define the measures ${\boldsymbol{\nu}}^{*}_{n,r_{n}}$, $n\geq 1$, on
$\ell_{0}(\mathbb{R}^{d})$ as follows:

$\displaystyle{\boldsymbol{\nu}}^{*}_{n,r_{n}}$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}\left[\delta_{u_{n}^{-1}\boldsymbol{X}_{1,r_{n}}}\right]\;.$

We are interested in convergence of ${\boldsymbol{\nu}}^{*}_{n,r_{n}}$ to
${\boldsymbol{\nu}}^{*}$. The results of this section are extracted from
[KS20, Chapter 6]. See also [PS18] and [BPS18].

####  2.6.1 Anticlustering conditions

For each fixed $r\in\mathbb{N}$, the distribution of
$u_{n}^{-1}\boldsymbol{X}_{-r,r}$ conditionally on
$\left|\boldsymbol{X}_{0}\right|>u_{n}$ converges weakly to the distribution
of $\boldsymbol{Y}_{-r,r}$ (see Section 2.3). In order to let $r$ tend to
infinity, we must embed all these finite vectors into one space of sequences.
By adding zeroes on each side of the vectors $u_{n}^{-1}\boldsymbol{X}_{-r,r}$
and $\boldsymbol{Y}_{-r,r}$ we identify them with elements of the space
$\ell_{0}(\mathbb{R}^{d})$. Then $\boldsymbol{Y}_{-r,r}$ converges (as
$r\to\infty$) to $\boldsymbol{Y}$ in $\ell_{0}(\mathbb{R}^{d})$ if (and only
if) $\boldsymbol{Y}\in\ell_{0}(\mathbb{R}^{d})$ almost surely.

However, this is not enough for statistical purposes and we consider the
following definition.

######  Definition 2.5 ([DH95], Condition 2.8).

Condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds if for all $x,y>0$,

$\displaystyle\lim_{m\to\infty}\limsup_{n\to\infty}\mathbb{P}\left(\max_{m\leq|j|\leq
r_{n}}|\boldsymbol{X}_{j}|>u_{n}x\mid|\boldsymbol{X}_{0}|>u_{n}y\right)=0\;.$
($\mathcal{A}\mathcal{C}(r_{n},u_{n})$)

Condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is referred to as the
anticlustering condition. It holds for i.i.d. regularly varying sequences if
$\lim_{n\to\infty}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})=0$.
Note that the latter condition is a part of the 1 assumption. It is also
fulfilled by many models, including geometrically ergodic Markov chains,
short-memory linear or max-stable processes.
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$ implies that
$\boldsymbol{Y}\in\ell_{0}(\mathbb{R}^{d})$. See [KSW19] and [KS20].

A stronger version of the anticlustering condition reads as follows.

######  Definition 2.6.

Condition $\mathcal{S}(r_{n},u_{n})$ holds if for all $s,t>0$

$\displaystyle\lim_{m\to\infty}\limsup_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=m}^{r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}s,\left|\boldsymbol{X}_{j}\right|>u_{n}t)=0\;.$
($\mathcal{S}(r_{n},u_{n})$)

The main consequence of the anticlustering condition
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is the following result.

######  Proposition 2.7 ([BS09], Proposition 4.2; [KS20], Theorem 6.1.4).

Let $H\in\mathcal{L}$. If Condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$
holds, then

$\displaystyle\lim_{n\to\infty}\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{-r_{n},r_{n}})\mid\left|\boldsymbol{X}_{0}\right|>u_{n}]=\mathbb{E}[H(\boldsymbol{Y})]\;.$

####  2.6.2 Vague convergence of cluster measure

We now state the unconditional convergence of
$u_{n}^{-1}\boldsymbol{X}_{1,r_{n}}$. Contrary to Proposition 2.7, where an
extreme value was imposed at time 0, a large value in the cluster can happen
at any time. Moreover, the convergence of
${\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)$ to $\boldsymbol{\nu}^{*}(H)$ may hold
only for shift-invariant functionals $H$. Therefore, we need the following
definition.

######  Definition 2.8.

The space $\tilde{\ell}_{0}(\mathbb{R}^{d})$ is the space of equivalence
classes of $\ell_{0}(\mathbb{R}^{d})$ endowed with the equivalence relation
$\sim$ defined by

$\displaystyle\boldsymbol{x}\sim\boldsymbol{y}\Longleftrightarrow\exists
j\in\mathbb{Z}\;,\ B^{j}\boldsymbol{x}=\boldsymbol{y}\;.$

######  Proposition 2.9 (Theorem 6.2.5 in [KS20]).

Let condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ hold. The sequence of
measures ${\boldsymbol{\nu}}^{*}_{n,r_{n}}$, $n\geq 1$ converges vaguely# on
$\tilde{\ell}_{0}(\mathbb{R}^{d})\setminus\\{{\boldsymbol{0}}\\}$ to
${\boldsymbol{\nu}}^{*}$, that is, for all $H\in\mathcal{A}$,

$\displaystyle\lim_{n\to\infty}{\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)=\lim_{n\to\infty}\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}=\boldsymbol{\nu}^{*}(H)\;.$

The immediate consequence is the following limit (cf. (2.2)):

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{P}(\boldsymbol{X}_{1,r_{n}}^{*}>u_{n})}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}={\vartheta}\;.$

####  2.6.3 Indicator functionals not vanishing around zero

Proposition 2.9 entails convergence of ${\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)$
for $H\in\mathcal{A}$. For functionals which are not defined on the whole
space $\ell_{0}(\mathbb{R}^{d})$ we need an additional assumption on
Asymptotic Negligibility of Small Jumps.

######  Definition 2.10.

Condition ${\rm ANSJB}(r_{n},u_{n})$ holds if for all $\eta>0$,

$\displaystyle\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{\mathbb{P}(\sum_{j=1}^{r_{n}}|\boldsymbol{X}_{j}|\mathbbm{1}{\left\\{|\boldsymbol{X}_{j}|\leq\epsilon
u_{n}\right\\}}>\eta
u_{n})}{r_{n}\mathbb{P}(|\boldsymbol{X}_{0}|>u_{n})}=0\;.$ (${\rm
ANSJB}(r_{n},u_{n})$)

######  Proposition 2.11 (Theorem 6.2.16 in [KS20]).

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ and ${\rm
ANSJB}(r_{n},u_{n})$ hold. Then for $K\in{\mathcal{K}}$,

$\displaystyle{\boldsymbol{\nu}}^{*}(\mathbbm{1}{\left\\{K>1\right\\}})=\lim_{n\to\infty}\frac{\mathbb{P}(K(\boldsymbol{X}_{1,r_{n}}/u_{n})>1)}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}={\vartheta}\int_{0}^{\infty}\mathbb{P}(K(z\boldsymbol{Q})>1)\alpha
z^{-\alpha-1}\mathrm{d}z<\infty\;.$

##  3 Central limit theorem for runs estimators

In this section we introduce and study runs estimators of cluster indices. A
pseudo-estimator is defined in (3.4). Its limiting covariance (for different
anchoring maps) is studied in Lemma 3.2. In particular, for two different
anchoring maps, the runs statistics are totally dependent. As a consequence we
cannot reduce the limiting variance for the estimation of
${\boldsymbol{\nu}}^{*}(H)$ by considering linear combinations of the runs
statistics. In Lemma 3.3 we consider covariance between runs and disjoint
blocks estimators. Again, we obtain total dependence in the limit. The main
result of the paper is the central limit theorem for runs estimators; see
Theorem 3.5. The limiting variance agrees with the one for the disjoint blocks
and sliding blocks estimators.

###  3.1 Runs estimator

To introduce runs estimators recall that (cf. (2.1))

$\displaystyle
H^{\mathcal{A}}\left(\boldsymbol{X}_{(j-1)r_{n}+h,(j+1)r_{n}+h}/u_{n}\right)=H^{\mathcal{A}}(B^{-h-jr_{n}}\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})$
$\displaystyle=H(B^{-h-jr_{n}}\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})\mathbbm{1}{\left\\{\mathcal{A}(B^{-h-jr_{n}}\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})=0\right\\}}\mathbbm{1}{\left\\{\left|B^{-h-jr_{n}}\boldsymbol{X}_{0}\right|>u_{n}\right\\}}$
$\displaystyle=H\left(\boldsymbol{X}_{(j-1)r_{n}+h,(j+1)r_{n}+h}/u_{n}\right)\times$
$\displaystyle\phantom{=}\ \ \ \
\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{X}_{(j-1)r_{n}+h,(j+1)r_{n}+h}/u_{n})=h+jr_{n}\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{h+jr_{n}}\right|>u_{n}\right\\}}\;.$
(3.1)

Set $q_{n}=n-r_{n}$ and $m_{n}=n/r_{n}$. Without loss of generality assume
that $m_{n}$ is an integer. Consider disjoint blocks

$\displaystyle J_{j}:=\\{jr_{n}+1,\ldots,(j+1)r_{n}\\}\;,\ \
j=0,\ldots,m_{n}-1\;.$ (3.2)

The union of these blocks gives $\\{1,\ldots,n\\}$. We assume we have data
$\boldsymbol{X}_{1-r_{n}},\ldots,\boldsymbol{X}_{n+r_{n}}$. For
$j=0,\ldots,m_{n}-1$ define

$\displaystyle H_{n,j}^{\mathcal{A}}$
$\displaystyle=\sum_{i=jr_{n}+1}^{(j+1)r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=\sum_{i=jr_{n}+1}^{(j+1)r_{n}}H^{\mathcal{A}}\left(B^{-i}\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)$
$\displaystyle=\sum_{i=jr_{n}+1}^{(j+1)r_{n}}H\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=i\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{i}\right|>u_{n}\right\\}}\;.$
(3.3)

Each $H_{n,j}^{\mathcal{A}}$ is a function of the block
$\boldsymbol{X}_{(j-1)r_{n}+1,\ldots,(j+2)r_{n}}$ of size $3r_{n}$. The number
$j$ in the notation $H_{n,j}^{\mathcal{A}}$ indicates that the indicator
$\left|\boldsymbol{X}_{i}\right|>u_{n}$ is applied with $i\in J_{j}$.

We consider a random process

$\displaystyle{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})=\frac{1}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)$
(3.4)

that can be decomposed as

$\displaystyle{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})=\frac{1}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=0}^{m_{n}-1}H_{n,j}^{\mathcal{A}}\;.$

If the anticlustering condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds,
then using stationarity, definition (2.6) of $H^{\mathcal{A}}$, Proposition
2.7 and (2.7) we have

$\displaystyle\lim_{n\to\infty}\mathbb{E}[{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})]=\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}\left[H^{\mathcal{A}}(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})\right]=\mathbb{E}[H^{\mathcal{A}}(\boldsymbol{Y})]={\boldsymbol{\nu}}^{*}(H)\;.$

Now, let $k_{n}$ be a sequence of integers (depending on $n$) such that
$k_{n}\to\infty$ and $k_{n}/n\to 0$. Define $u_{n}$ by
$k_{n}=n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$ and replace $u_{n}$
in $H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)$ with
$(k_{n}+1)$th order statistics ${\left|\boldsymbol{X}\right|}_{(n:n-k_{n})}$
to get the runs estimator:

$\displaystyle\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})=\frac{1}{k_{n}}\sum_{i=1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/{\left|\boldsymbol{X}\right|}_{(n:n-k_{n})}\right)\;.$
(3.5)

In what follows we will use interchangeably $k_{n}$ and
$n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$, whatever is more
suitable.

###  3.2 Mixing assumptions

Dependence in $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ will be controlled by
the $\beta$-mixing rates $\\{\beta_{n}\\}$. Recall 1. Let $\\{\ell_{n}\\}$ be
a sequence of integers such that $\lim_{n\to\infty}\ell_{n}=\infty$ and
$\lim_{n\to\infty}\ell_{n}/r_{n}=0$.

######  Definition 3.1.

Condition $\beta^{\prime}(r_{n})$ holds if:

$\displaystyle\lim_{n\to\infty}\frac{n}{r_{n}}\beta_{r_{n}}=0\;,$ (3.6a)
$\displaystyle\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=r_{n}+1}^{\infty}\beta_{i}=\lim_{n\to\infty}\frac{n}{k_{n}}\sum_{i=r_{n}+1}^{\infty}\beta_{i}=0\;,$
(3.6b)
$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{\infty}\beta_{jr_{n}}=\lim_{n\to\infty}\frac{n}{r_{n}k_{n}}\sum_{j=1}^{\infty}\beta_{jr_{n}}=0\;.$
(3.6c)

###  3.3 Limiting covariances

####  3.3.1 Runs statistics

The first result deals with covariance of the process
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}$ defined in (3.4).

######  Lemma 3.2.

Assume 1, $\mathcal{A}\mathcal{C}(r_{n},u_{n})$, $\mathcal{S}(r_{n},u_{n})$
and (3.6b) hold. Let $H,\widetilde{H}\in\mathcal{L}$ and
$\mathcal{A},\widetilde{\mathcal{A}}$ be anchoring maps. Then

$\displaystyle\lim_{n\to\infty}n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})\mathrm{cov}\left({\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}}),{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(\widetilde{H}^{\widetilde{\mathcal{A}}})\right)={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$
(3.7)

We note that the limit does not depend on the choice of the anchoring maps. In
other words, for two different anchoring maps, $\mathcal{A}$ and
$\widetilde{\mathcal{A}}$, the runs statistics
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$ and
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\widetilde{\mathcal{A}}})$
are totally dependent. As a consequence we cannot reduce the limiting variance
for the estimation of ${\boldsymbol{\nu}}^{*}(H)$ by considering a linear
combination of ${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$
and
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\widetilde{\mathcal{A}}})$.

####  3.3.2 Runs and disjoint blocks statistics

We analyse covariance between
${\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$ defined in
(3.4) and the disjoint blocks statistics

$\displaystyle\frac{1}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=0}^{m_{n}-1}H\left(\boldsymbol{X}_{jr_{n}+1,(j+1)r_{n}}/u_{n}\right)={\widetilde{\boldsymbol{\nu}}}^{*}_{n,r_{n}}(H)\;.$
(3.8)

The disjoint blocks statistics are considered in [KS20, Chapter 10].

######  Lemma 3.3.

Assume 1, $\mathcal{A}\mathcal{C}(r_{n},u_{n})$, $\mathcal{S}(r_{n},u_{n})$
and (3.6c) hold. Let $H,\widetilde{H}\in\mathcal{L}$,
$\widetilde{H}({\boldsymbol{0}})=0$ and $\mathcal{A}$ be an anchoring map.
Then

$\displaystyle\lim_{n\to\infty}n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})\mathrm{cov}\left({\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}}),{\widetilde{\boldsymbol{\nu}}}^{*}_{n,r_{n}}(\widetilde{H})\right)={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$
(3.9)

Again, irrespectively of the choice of the anchoring map $\mathcal{A}$, the
runs and disjoint blocks statistics are totally dependent and we cannot reduce
the limiting variance by considering their linear combinations.

###  3.4 Central limit theorem

Let ${\mathbb{G}}$ be the Gaussian process on $L^{2}({\boldsymbol{\nu}}^{*})$
with covariance

$\displaystyle\mathrm{cov}({\mathbb{G}}(H),{\mathbb{G}}(\widetilde{H}))={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$

Recall that for a functional
$H:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}_{+}$ and $s>0$ we define
$H_{s}(\boldsymbol{x})=H(\boldsymbol{x}/s)$. Also, recall that
$\mathcal{E}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{|\boldsymbol{x}_{j}|>1\right\\}}$.

Consider the class

$\mathcal{G}=\\{H^{\mathcal{A}}_{s},s\in[s_{0},t_{0}]\\}=\\{H(\boldsymbol{x}/s)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x}/s)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}},\;\;s\in[s_{0},t_{0}]\\}.$

We need the following assumption on its random entropy.

######  Assumption 3.4.

There exists a random metric $d_{n}$ on $\mathcal{G}$ and a measurable
majorant $N^{*}(\mathcal{G},d_{n},\epsilon)$ of the covering number
$N(\mathcal{G},d_{n},\epsilon)$ such that for every sequence
$\\{\delta_{n}\\}$ which decreases to zero,

$\displaystyle\int_{0}^{\delta_{n}}\sqrt{\log
N^{*}(\mathcal{G},d_{n},\epsilon)}\mathrm{d}\epsilon\stackrel{{\scriptstyle\tiny\mathbb{P}}}{{\longrightarrow}}0\;.$
(3.10)

The main result of this paper is Theorem 3.5, the asymptotic normality of the
appropriately normalized estimator
$\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$. The limiting
variance agrees with the one for the disjoint blocks and sliding blocks
estimators; cf. [DR10], [KS20, Chapter 10], [CK21].

######  Theorem 3.5.

Let $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ be a stationary, regularly
varying $\mathbb{R}^{d}$-valued time series. Assume that 1,
$\beta^{\prime}(r_{n})$, $\mathcal{S}(r_{n},u_{n})$ and

$\displaystyle\lim_{n\to\infty}\frac{r_{n}}{\sqrt{k_{n}}}=\lim_{n\to\infty}\frac{r_{n}}{\sqrt{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}}=0\;$
(3.11)

hold. Suppose that Assumption 3.4 is satisfied. Fix $0<s_{0}<1<t_{0}<\infty$.
Assume moreover that for
$\mathcal{A}=\mathcal{A}^{(0)},\mathcal{A}^{(1)},\mathcal{A}^{(2)}$,

$\displaystyle\lim_{n\to\infty}\sqrt{k_{n}}\sup_{s\in[s_{0},t_{0}]}\left|\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}s)}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}-s^{-\alpha}\right|$
$\displaystyle=0\;,$ (3.12a)
$\displaystyle\lim_{n\to\infty}\sqrt{k_{n}}\sup_{s\in[s_{0},t_{0}]}|\mathbb{E}[{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H_{s}^{\mathcal{A}})]-{\boldsymbol{\nu}}^{*}(H_{s})|$
$\displaystyle=0\;.$ (3.12b)

If $H\in\mathcal{A}$, then

$\displaystyle\sqrt{k_{n}}\left\\{\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})-{\boldsymbol{\nu}}^{*}(H)\right\\}\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}{\mathbb{G}}(H-{\boldsymbol{\nu}}^{*}(H)\mathcal{E})\;.$
(3.13)

If moreover ${\rm ANSJB}(r_{n},u_{n})$ is satisfied, then (3.13) holds for
$H\in\mathcal{B}$.

####  3.4.1 Comments on the conditions

One chooses typically $k_{n}=n^{\epsilon}$ with some $\epsilon\in(0,1)$. We
note that $\beta^{\prime}(r_{n})$ holds if e.g. $\beta_{n}=O(n^{-\delta})$
with $\delta>1$ big enough or if $\beta_{n}$ decays logarithmically. In the
latter case, we typically choose $r_{n}=(\log n)^{1+\delta}$ with some
$\delta>0$. Recalling the choice of $k_{n}$ we can see that (3.11) is not a
very stringent assumption.

Furthermore, (3.12a) controls the bias in the tail empirical process and can
be related to the classical second order assumptions.

Assumption 3.4 controls the size of the class $\mathcal{G}$. We are not able
to provide a general set of conditions under which this condition is
satisfied, however, we will verify it for virtually all functionals $H$ that
appeared in the paper. See Section 5.8.

##  4 Simulation study

We conducted some simulations in order to study the finite sample performance
of the runs estimators for selected cluster indices. We compare their
performance with the disjoint and sliding blocks estimators (see [CK21]).
Recall that the limiting variances are the same for all estimators. We do not
have theoretical formulas for bias. We note that the bias for disjoint and
sliding blocks estimators is the same, but different for runs estimators. We
present only a small portion of our simulation studies; the most important
findings are summarized at the end of the section.

###  4.1 Stationary AR process

We start with a simple AR(1) process. For this process we have explicit
formulas for all cluster indices. Samples of size $n=1000$ are generated from
AR(1) with $\alpha=4$ and $\rho=0.5,0.9$. We perform simulations for the
classical extremal index as well as for the stop-loss index.

##### Extremal index.

For AR(1) with $\rho>0$ the extremal index is $\theta=1-\rho^{\alpha}$; cf.
[KS20, p. 396].

  * •

Table 1 includes the results for Monte Carlo simulation for the extremal index
based on disjoint blocks, sliding blocks and runs estimators with the block
size $r_{n}=8,\;9$. We used $k=5\%$ and $10\%$ order statistics. We note that
for the strong dependence ($\rho=0.9$), the sliding and disjoint blocks
estimators outperform runs estimators for all considered parameters. For weak
dependence ($\rho=0.5$), the results are heavily biased for all considered
parameters. We note that all estimators yield almost the same variances, which
is in agreement with the theoretical results obtained in the paper.

We note that the fact that stronger dependence yields smaller variability of
the estimators is not surprising, cf. e.g. Figure 5 in [RSF09].

##### Stop-loss index.

For AR(1) with $\rho>0$ the formula for the stop-loss index is given in [KS20,
p. 619]:

$\displaystyle\theta_{{\rm stop-
loss}}(S)=(1-\rho^{\alpha})\mathbb{P}\left(\sum_{j=0}^{\infty}(\rho^{j}Y_{0}-1)_{+}>S\right)\;,$
(4.1)

where $Y_{0}$ is a Pareto random variable with the parameter $\alpha$.

  * •

At the first step we use the formula (4.1) and performed the Monte-Carlo
simulation to obtain the approximate value of the stop-loss index.

  * •

With this in mind, we performed simulation studies for $k=10\%$ and $k=40\%$.
As noted in [CK21], the stop-loss index estimation requires a higher number of
order statistics. We notice then (see Table 2) that, as opposed to the
extremal index, the runs estimator with the anchoring map $C^{(0)}$ performs
better than the disjoint and the sliding blocks in case of weaker dependence.
Indeed, the weaker dependence ($\rho=0.5$) yields a good estimation for runs
estimators for any given block size, while for the strong dependence
($\rho=0.9$), the simulation results are rather poor for all the estimators.
This may be quite intuitive, since the stop-loss functional is based on sums
of large values. On the other side, all the estimators perform poorly for
strong dependence ($\rho=0.9$) as a result of bias.

  * •

The box plots in Figure 1 is based again on Monte Carlo simulations. The
following parameters are used: $\rho=0.5$, $\alpha=4$ and the block size
$r_{n}=8,\;9$ along with $k=10\%$ and $k=40\%$. We notice again that in both
cases the runs estimator yields acceptable result as opposed to disjoint and
sliding blocks estimators.

###  4.2 Stationary ARCH process

We consider a stationary ARCH(1) process defined by
$X_{j}^{2}=\sqrt{\beta+\lambda X_{j-1}^{2}}Z_{j}$, where
$\\{Z_{j},j\in\mathbb{Z}\\}$ are i.i.d standard normal random variables. For
$\lambda=0.9$ the extremal index is $\theta=0.612$ (see [EKM97, p. 480]).

  * •

Monte Carlo results are included in Table 3. In this case both disjoint and
sliding blocks estimators yield better results as compared to runs. This is
primarily due to bias.

In summary,

  * •

All estimators (blocks and runs) have the same variance, which is in line with
the theoretical results.

  * •

For the extremal index, runs estimators are inferior as compared to blocks
estimators. This is primarily due to bias.

  * •

For the stop-loss index, runs estimators are superior, yielding lower
(simulated) bias as compared to the blocks estimators.

##  5 Proofs

In Section 5.2 we prove several lemmas on conditional convergence when
anchoring maps are involved. One needs to distinguish between finite blocks
(when the conditional convergence follows basically from the conditional
convergence to the tail process) and growing blocks (when the anticlustering
condition is needed).

In Section 5.3 we prove the asymptotic behaviour of the covariances of runs
estimators, that is we prove Lemma 3.2 and Lemma 3.3. Section 5.4 deals with
the empirical cluster process of runs statistics. The functional central limit
theorem (Theorem 5.5) established there yields immediately the central limit
theorem for runs estimators. See Section 5.5. A long proof of Theorem 5.5 is
given in Sections 5.6 and 5.7. Finally, in Section 5.8 we discuss the random
entropy assumption.

###  5.1 Mixing

We recall the covariance inequality for bounded, beta-mixing random variables
(in fact, the inequality holds for $\alpha$-mixing). Let
$\beta({\mathcal{F}_{1}},{\mathcal{F}_{2}})$ be the $\beta$-mixing coefficient
between two sigma fields. Then (cf. [Ibr62])

$\displaystyle|\mathrm{cov}(H(Z_{1}),H(Z_{2}))|\leq\mathrm{cst}\
\|H\|_{\infty}\|\widetilde{H}\|_{\infty}\beta(\sigma(Z_{1}),\sigma(Z_{2}))\;.$
(5.1)

In (5.1) the constant $\mathrm{cst}$ does not depend on $H,\widetilde{H}$.

###  5.2 Conditional convergence

######  Lemma 5.1.

Let $\mathcal{A}=\mathcal{A}^{(0)},\mathcal{A}^{(1)},\mathcal{A}^{(2)}$. Then
for $h\ \in\mathbb{Z}$,

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})=\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r,h+r})=h)\;.$

###### Proof.

We will do the proof for $\mathcal{A}^{(0)}$ only. By the definition of the
tail process

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}^{(0)}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}(\boldsymbol{X}_{h-r,h-1}^{*}/u_{n}<\left|\boldsymbol{X}_{h}\right|/u_{n},\boldsymbol{X}_{h+1,h+r}^{*}/u_{n}\leq\left|\boldsymbol{X}_{h}\right|/u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\boldsymbol{Y}_{h-r,h-1}^{*}<\left|\boldsymbol{Y}_{h}\right|,\boldsymbol{Y}_{h+1,h+r}^{*}\leq\left|\boldsymbol{Y}_{h}\right|)=\mathbb{P}(\mathcal{A}^{(0)}(\boldsymbol{Y}_{h-r,h+r})=h)\;.$

∎

######  Lemma 5.2.

Let $\mathcal{A},\widetilde{\mathcal{A}}$ be any of the anchoring maps
$\mathcal{A}^{(0)},\mathcal{A}^{(1)},\mathcal{A}^{(2)}$. Then for $h\
\in\mathbb{Z}$,

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{-r,r}/u_{n})=0,\widetilde{\mathcal{A}}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r,r})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r,h+r})=h)\;.$
(5.2)

###### Proof.

We verify the statement for one combination of the anchoring maps only. For
$\mathcal{A}^{(1)}$ and $\mathcal{A}^{(2)}$ , we have

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}^{(1)}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h,\mathcal{A}^{(2)}(\boldsymbol{X}_{-r,r}/u_{n})=0\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\boldsymbol{X}_{h-r,h-1}^{*}\leq
u_{n},\left|\boldsymbol{X}_{h}\right|>u_{n},\boldsymbol{X}_{1,r}^{*}\leq
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\boldsymbol{Y}_{h-r,h-1}^{*}\leq
1,\left|\boldsymbol{Y}_{h}\right|>1,\boldsymbol{Y}_{1,r}^{*}\leq
1,\left|\boldsymbol{Y}_{0}\right|>1)\;.$
$\displaystyle=\mathbb{P}(\mathcal{A}^{(1)}(\boldsymbol{Y}_{h-r,h+r})=h,\mathcal{A}^{(2)}(\boldsymbol{Y}_{-r,r})=0)\;.$

∎

Recall the definition of $H^{\mathcal{A}}$ in (3.1). Let $h_{n}$ be a sequence
of integers diverging to infinity. For bounded $H,\widetilde{H}$, a direct
application of $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ gives

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{h_{n}-r_{n},h_{n}+r_{n}}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle\leq\|H\|\|\widetilde{H}\|\lim_{n\to\infty}\mathbb{P}(\left|\boldsymbol{X}_{h_{n}}\right|>u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})=0\;.$

Likewise, if $H,\widetilde{H}$ are bounded and $\widetilde{H}\in\mathcal{L}$
is such that $\widetilde{H}({\boldsymbol{0}})=0$, then the Lipschitz
continuity of $\widetilde{H}$ and $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ imply

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{h_{n},h_{n}+r_{n}}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle\leq\|H\|\lim_{n\to\infty}\mathbb{E}\left[\widetilde{H}\left(\boldsymbol{X}_{h_{n},h_{n}+r_{n}}/u_{n}\right)-\widetilde{H}({\boldsymbol{0}})\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle\leq\|H\|\|\widetilde{H}\|\lim_{n\to\infty}\mathbb{P}\left(\boldsymbol{X}_{h_{n},h_{n}+r_{n}}^{*}>u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right)=0\;.$
(5.3)

The statement in (5.2) is also valid if
$\widetilde{H}\left(\boldsymbol{X}_{h_{n},h_{n}+r_{n}}/u_{n}\right)$ is
replaced with
$\widetilde{H}\left(\boldsymbol{X}_{-h_{n},-h_{n}-r_{n}}/u_{n}\right)$.

On the other hand, for fixed $h$ we have the following lemma that extends
Lemma 5.2 from fixed $r$ to $r_{n}\to\infty$. For this, we need to assume
additionally that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds.

######  Lemma 5.3.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let
$H,\widetilde{H}\in\mathcal{L}$ and $\mathcal{A},\widetilde{\mathcal{A}}$ be
anchoring maps. Then

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle=\mathbb{E}\left[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\widetilde{\mathcal{A}}(\boldsymbol{Y})=h\right\\}}\right]=:{\mathcal{I}}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}};h)\;.$
(5.4)

Before we prove the above lemma, we make several comments.

First, as a corollary we obtain

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]={\boldsymbol{\nu}}^{*}(H)\;$
(5.5)

and

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})=0\mid\left|\boldsymbol{X}_{0}\right|>u_{n})={\boldsymbol{\nu}}^{*}(1)=1\;.$
(5.6)

Indeed, if we take $\widetilde{H}\equiv 1$,
$\mathcal{A}=\widetilde{\mathcal{A}}$ and $h=0$, then by (2.7),

$\displaystyle{\mathcal{I}}(H,1,\mathcal{A},{\mathcal{A}};0)$
$\displaystyle=\mathbb{E}\left[H(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\right]$
$\displaystyle=\mathbb{E}\left[H(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{Y}_{0}\right|>1\right\\}}\right]={\boldsymbol{\nu}}^{*}(H)\;.$

Since the definition of ${\boldsymbol{\nu}}^{*}$ does not depend on the
anchoring map, we have
${\mathcal{I}}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}};0)={\boldsymbol{\nu}}^{*}(H\widetilde{H})$
for any $\mathcal{A},\widetilde{\mathcal{A}}$. Since the value of any
anchoring map is uniquely determined, we conclude immediately that
${\mathcal{I}}(H,\widetilde{H},\mathcal{A},{\mathcal{A}};h)=0$ for $h\not=0$.
Furthermore,

$\displaystyle\sum_{h\in\mathbb{Z}}\mathbb{E}\left[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\widetilde{\mathcal{A}}(\boldsymbol{Y})=h\right\\}}\right]$
$\displaystyle=\mathbb{E}\left[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\widetilde{\mathcal{A}}(\boldsymbol{Y})\in\mathbb{Z}\right\\}}\right]={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$

This implies that for arbitrary anchoring maps $\mathcal{A}$,
$\widetilde{\mathcal{A}}$,

$\displaystyle{\mathcal{I}}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}};h)=0\;,\
\ h\not=0\;.$ (5.7)

###### Proof of Lemma 5.3.

In [CK21] we proved a version of the lemma without anchoring maps included.
Since
$\boldsymbol{x}\to\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x})=0\right\\}}$
is not Lipschitz continuous, Lemma 6.6 in [CK21] is not directly applicable.
As such, we will focus on the anchoring maps only, assuming
$H=\widetilde{H}\equiv 1$.

In the first step we prove that for all $h\in\mathbb{Z}$

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\mathcal{A}(\boldsymbol{Y})=h)\;.$ (5.8)

We already know that (cf. Lemma 5.1)

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})=\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r,h+r})=h)\;.$

Since $r_{n}\to\infty$ and $r$ is fixed we can assume $0<r<r_{n}$. Now, for
$\mathcal{A}=\mathcal{A}^{(0)},\mathcal{A}^{(1)},\mathcal{A}^{(2)}$ the value
of

$\displaystyle\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r,h+r})=h)-\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)$

is non zero if and only if
$h+r<\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})\leq h+r_{n}$ or
$h-r_{n}\leq\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})<h-r$. Indeed, take
for simplicity $h=0$. If $\mathcal{A}^{(1)}(\boldsymbol{Y}_{-r,r})=0$ and
$\mathcal{A}^{(1)}(\boldsymbol{Y}_{-r_{n},r_{n}})\not=0$, then
$\boldsymbol{Y}_{-r,-1}^{*}\leq 1$, $\left|\boldsymbol{Y}_{0}\right|>1$ and
then $\boldsymbol{Y}_{-r_{n},-r-1}^{*}>1$, while
$\mathcal{A}^{(1)}(\boldsymbol{Y}_{-r,r})\not=0$ and
$\mathcal{A}^{(1)}(\boldsymbol{Y}_{-r_{n},r_{n}})=0$ cannot happen. The same
reasoning applied to the other anchoring maps.

Coming back to the general case of $h$, the first property of the anchoring
map implies that $\left|\boldsymbol{Y}_{j}\right|>1$ for some
$j\in\\{h+r+1,\ldots,h+r_{n}\\}\cup\\{h-r_{n},\ldots,h-r-1\\}$. Since we let
$r,r_{n}\to\infty$, we can assume that $h<r$. Thus, using the property An(i)
of the anchoring maps,

$\displaystyle\lim_{r\to\infty}{\lim_{n\to\infty}}|\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r,h+r})=h)-\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)|$
$\displaystyle\leq\lim_{r\to\infty}{\lim_{n\to\infty}}\mathbb{P}\left(\max\left\\{\max_{h+r\leq
j\leq h+r_{n}}\left|\boldsymbol{Y}_{j}\right|,\max_{h-r_{n}\leq j\leq
h-r}\left|\boldsymbol{Y}_{j}\right|\right\\}>1\right)=0$ (5.9)

since $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ implies
$\boldsymbol{Y}_{j}\to{\boldsymbol{0}}$ almost surely as $|j|\to\infty$. Also,
the vanishing property of $\boldsymbol{Y}_{j}$ and the property An(i) of the
anchoring map imply that

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)=\mathbb{P}(\mathcal{A}(\boldsymbol{Y})=h)\;.$

Similarly,

$\displaystyle\left|\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})-\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\right|$

is non zero if and only if
$h+r<\mathcal{A}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})\leq h+r_{n}$ or
$h-r_{n}\leq\mathcal{A}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})<h-r$. The
first property of the anchoring map implies that
$\left|\boldsymbol{X}_{j}\right|>\left|\boldsymbol{X}_{0}\right|\wedge u_{n}$
for some $j\in\\{h+r+1,\ldots,h+r_{n}\\}\cup\\{h-r_{n},\ldots,h-r-1\\}$.
Again, we can assume that $h<r$. Keeping in mind the conditioning we have:

$\displaystyle\lim_{r\to\infty}\lim_{n\to\infty}\left|\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})-\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{h-r,h+r}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\right|$
$\displaystyle\leq\lim_{r\to\infty}\lim_{n\to\infty}\mathbb{P}\left(\max\left\\{\max_{h+r\leq
j\leq h+r_{n}}\left|\boldsymbol{X}_{j}\right|,\max_{h-r_{n}\leq j\leq
h-r}\left|\boldsymbol{X}_{j}\right|\right\\}>u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right)=0$
(5.10)

by $\mathcal{A}\mathcal{C}(r_{n},u_{n})$. This finishes the proof of (5.8).

Now, we will prove

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\mathcal{A}(\boldsymbol{X}_{-r_{n},+r_{n}}/u_{n})=0,\widetilde{\mathcal{A}}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})=h\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\mathbb{P}(\mathcal{A}(\boldsymbol{Y})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y})=h)\;.$
(5.11)

In view of Lemma 5.2, (5.2) holds with $r_{n}$ replaced with $r$. Now, the
idea is to reduce the bivariate case to the univariate.

Note first that for the anchoring maps considered here, the event
$A_{1}:=\\{\mathcal{A}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h\\}$ is included in
$A_{2}:=\\{\mathcal{A}(\boldsymbol{Y}_{h-r,h+r})=h\\}$. We also note that for
any event $B$ and any pair of ordered events $A_{1},A_{2}$ we have

$|\mathbb{P}(A_{1}\cap B)-\mathbb{P}(A_{2}\cap
B)|\leq|\mathbb{P}(A_{1})-\mathbb{P}(A_{2})|\;.$

Thus, we can bound

$\displaystyle\left|\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r,r})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r,h+r})=h)-\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r_{n},r_{n}})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)\right|$

by

$\displaystyle\left|\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r,r})=0)-\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r_{n},r_{n}})=0)\right|+\left|\mathbb{P}(\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r,h+r})=h)-\mathbb{P}(\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)\right|$

and we use the first step to conclude that

$\displaystyle\lim_{r\to\infty}\lim_{n\to\infty}\left|\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r,r})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r,h+r})=h)-\mathbb{P}(\mathcal{A}(\boldsymbol{Y}_{-r_{n},r_{n}})=0,\widetilde{\mathcal{A}}(\boldsymbol{Y}_{h-r_{n},h+r_{n}})=h)\right|=0\;.$

Therefore, (5.2) can be extended to the bivariate case. The same argument
allows to extend (5.2) to the bivariate case. In summary, the proof of (5.2)
is finished. ∎

In the next lemma, we analyse the conditional convergence for the product of
$H^{\mathcal{A}}$ and $\widetilde{H}$. Its proof is almost the same as above
and hence it is omitted.

######  Lemma 5.4.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let
$H,\widetilde{H}\in\mathcal{L}$, $\widetilde{H}({\boldsymbol{0}})=0$ and
$\mathcal{A}$ be an anchoring map. Then, for $h,h^{\prime}\geq 0$,

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{h-r_{n},h^{\prime}+r_{n}}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle=\mathbb{E}\left[H\left(\boldsymbol{Y}\right)\widetilde{H}\left(\boldsymbol{Y}\right)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\right]={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$
(5.12)

###  5.3 Limiting Covariances

The goal of this section is to prove Lemmas 3.2 and 3.3. Two situations will
arise when dealing with the covariances:

  * •

Situation 1: we will deal with
$\sum_{h=-r_{n}}^{r_{n}}\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]$, where

$\displaystyle\lim_{n\to\infty}\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]=\mathbb{E}[c_{h}(\boldsymbol{Y})]\;,\
\ \sum_{h\in\mathbb{Z}}\mathbb{E}[|c_{h}(\boldsymbol{Y})|]<\infty\;.$

We will fix an integer $r>0$; the convergence of
$\sum_{h=-r}^{r}\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]$ to
$\sum_{h=-r}^{r}\mathbb{E}[c_{h}(\boldsymbol{Y})]$ will follow. The reminder
$\sum_{|h|>r}\mathbb{E}[c_{h}(\boldsymbol{Y})]$ is negligible (as
$r\to\infty$) by the summability assumption, while
$\sum_{h>|r_{n}|}\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]$ will be treated by
the anticlustering condition $\mathcal{S}(r_{n},u_{n})$.

  * •

Situation 2: we will deal with
$r_{n}^{-1}\sum_{h=1}^{r_{n}}\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]=\int_{0}^{1}g_{n}(\xi)\mathrm{d}\xi$,
where $g_{n}(h)=\mathbb{E}[c_{h,n}(\boldsymbol{X}/u_{n})]$ and $g_{n}(\xi)\to
g(\xi)$ as $n\to\infty$. Bounded convergence argument will be applied.

###### Proof of Lemma 3.2.

Recall that

$\displaystyle H_{n,j}^{\mathcal{A}}$
$\displaystyle=\sum_{i=jr_{n}+1}^{(j+1)r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;.$

The covariance of the scaled statistics is

$\displaystyle
n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})\mathrm{cov}\left({\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}}),{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(\widetilde{H}^{\widetilde{\mathcal{A}}})\right)=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}\left(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,0}^{\widetilde{\mathcal{A}}}\right)$
$\displaystyle\phantom{=}+\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{m_{n}-1}\left(1-\frac{j}{m_{n}}\right)\left\\{\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,j}^{\widetilde{\mathcal{A}}})+\mathrm{cov}(\widetilde{H}_{n,0}^{\widetilde{\mathcal{A}}},H_{n,j}^{\mathcal{A}})\right\\}\;.$
(5.13)

With the help of 1, we will show that
$\mathrm{cov}\left(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,0}^{\widetilde{\mathcal{A}}}\right)$
is determined by that of

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}\left(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,0}^{\widetilde{\mathcal{A}}}\right)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=-r_{n}}^{r_{n}}\left(1-\frac{|h|}{r_{n}}\right)\mathbb{E}[H^{\mathcal{A}}\left(\boldsymbol{X}_{0,2r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{h,h+2r_{n}}/u_{n}\right)]$
$\displaystyle={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$ (5.14)

We are in the Situation 1. For fixed $r$, using (5.3) and (5.7) we have

$\displaystyle\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=-r}^{r}\mathbb{E}[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)]$
$\displaystyle=\sum_{h=-r}^{r}\mathbb{E}[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\widetilde{\mathcal{A}}(\boldsymbol{Y})=h\right\\}}]=\sum_{h=-r}^{r}\mathcal{I}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}},h)=$
$\displaystyle=\mathcal{I}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}},0)=\mathbb{E}[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}]={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$
(5.15)

The value above does not depend on $r$. Moreover,

$\displaystyle\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{r<|h|\leq
r_{n}}\mathbb{E}[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{-r_{n}+h,r_{n}+h}/u_{n}\right)]$
$\displaystyle\leq\|H\|\|\widetilde{H}\|\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{r<|h|\leq
r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{h}\right|>u_{n})\;.$
(5.16)

Letting $n\to\infty$ and then $r\to\infty$, we finish the proof of (5.3) by
applying $\mathcal{S}(r_{n},u_{n})$.

Now, we deal with the term in (5.3). For $j\geq 1$,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\left|\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,j}^{\widetilde{\mathcal{A}}})\right|$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\left|\mathrm{cov}\left(\sum_{h=1}^{r_{n}}H^{\mathcal{A}}\left(B^{-h}\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right),\sum_{i=1}^{r_{n}}\widetilde{H}^{\widetilde{\mathcal{A}}}\left(B^{-i}\boldsymbol{X}_{(j-1)r_{n},(j+1)r_{n}}/u_{n}\right)\right)\right|$
$\displaystyle\leq\sum_{h=(j-1)r_{n}+1}^{jr_{n}}\left(\frac{h}{r_{n}}-(j-1)\right)|{g}_{n}(h)|+\sum_{h=jr_{n}+1}^{(j+1)r_{n}}\left((j+1)-\frac{h}{r_{n}}\right)|{g}_{n}(h)|$
$\displaystyle\leq\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}|{g}_{n}(h)|=:I_{j}$
(5.17)

with

$\displaystyle{g}_{n}(h)$
$\displaystyle=\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}(H^{\mathcal{A}}(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}),\widetilde{H}^{\widetilde{\mathcal{A}}}(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}))\;.$

For $h>2r_{n}$ we have by (5.1),

$\displaystyle|{g}_{n}(h)|$
$\displaystyle\leq\frac{\|H\|_{\infty}\|\widetilde{H}\|_{\infty}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\beta_{h-2r_{n}}\;.$
(5.18)

Thus,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{m_{n}-1}\left|\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{n,j}^{\widetilde{\mathcal{A}}})\right|\leq\frac{\|H\|_{\infty}\|\widetilde{H}\|_{\infty}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{m_{n}-1}\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}\beta_{h-2r_{n}}$
$\displaystyle\leq
2\frac{\|H\|_{\infty}\|\widetilde{H}\|_{\infty}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=3r_{n}+1}^{\infty}\beta_{h-2r_{n}}=O(1)\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=r_{n}+1}^{\infty}\beta_{i}=o(1)$

by the assumption (3.6b).

The terms that correspond to $j=1,2,3$ in (5.3) have to be dealt with
separately. We are again in the Situation 1. We have

$\displaystyle I_{1}+I_{2}+I_{3}\leq
2\sum_{h=1}^{4r_{n}}|g_{n}(h)|=2\left\\{\sum_{h=1}^{r}+\sum_{i=r+1}^{4r_{n}}\right\\}|g_{n}(h)|\;.\;$

Both parts are negligible. Indeed, as in (5.3),

$\displaystyle\lim_{n\to\infty}\sum_{h=1}^{r}|g_{n}(h)|\leq\lim_{n\to\infty}\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=1}^{r}\mathbb{E}[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}^{\widetilde{\mathcal{A}}}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)]$
$\displaystyle=\sum_{h=1}^{r}\mathbb{E}[H(\boldsymbol{Y})\widetilde{H}(\boldsymbol{Y})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{Y})=0\right\\}}\mathbbm{1}{\left\\{\widetilde{\mathcal{A}}(\boldsymbol{Y})=h\right\\}}]=\sum_{h=1}^{r}\mathcal{I}(H,\widetilde{H},\mathcal{A},\widetilde{\mathcal{A}},h)$

and by (5.7) the last term vanishes.

For the term $\sum_{h=r+1}^{4r_{n}}$ we apply $\mathcal{S}(r_{n},u_{n})$; see
the argument used in (5.3).

This finishes the proof of the lemma. ∎

###### Proof of Lemma 3.3.

Recall that

$\displaystyle\widetilde{H}_{j}=H\left(\boldsymbol{X}_{jr_{n}+1,(j+1)r_{n}}/u_{n}\right)\;.$

Here, $\widetilde{H}_{j}$ is a function of the $j$th block
$\boldsymbol{X}_{jr_{n}+1,(j+1)r_{n}}$, $j=0,\ldots,m_{n}-1$. Since
$H_{n,j}^{\mathcal{A}}$, $j=0,\ldots,m_{n}-1$, is a function of the block
$\boldsymbol{X}_{(j-1)r_{n}+1,\ldots,(j+2)r_{n}}$ (recall that we assumed that
we have data $\boldsymbol{X}_{1-r_{n}},\ldots,\boldsymbol{X}_{n+r_{n}}$), for
$|q|\geq 3$,

$\displaystyle\mathrm{cov}(H_{n,j}^{\mathcal{A}},\widetilde{H}_{j+q})\leq\|H\|\|\widetilde{H}\|\beta_{(|q|-2)r_{n}}\;;$
(5.19)

cf. (5.1). We have

$\displaystyle
k_{n}\mathrm{cov}\left({\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}}),{\widetilde{\boldsymbol{\nu}}}^{*}_{n,r_{n}}(\widetilde{H})\right)=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}\left(H_{n,0}^{\mathcal{A}},\widetilde{H}_{0}\right)$
$\displaystyle\phantom{=}+\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{m_{n}-1}\left(1-\frac{j}{m_{n}}\right)\left\\{\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{j})+\mathrm{cov}(\widetilde{H}_{0},H_{n,j}^{\mathcal{A}})\right\\}\;.$
(5.20)

We analyse $\mathrm{cov}\left(H_{n,0}^{\mathcal{A}},\widetilde{H}_{0}\right)$.

We are in the Situation 2:

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}\left[H_{n,0}^{\mathcal{A}}\widetilde{H}_{0}\right]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{1,r_{n}}/u_{n}\right)\right]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}\right)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{i}\right|>u_{n}\right\\}}\widetilde{H}\left(\boldsymbol{X}_{1,r_{n}}/u_{n}\right)\right]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}\right)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{0}\right|>u_{n}\right\\}}\widetilde{H}\left(\boldsymbol{X}_{1-i,r_{n}-i}/u_{n}\right)\right]$
$\displaystyle=\frac{1}{r_{n}}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{1-i,r_{n}-i}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]=\int_{0}^{1}h_{n,0}(\xi)\mathrm{d}\xi$

with

$\displaystyle
h_{n,0}(\xi)=\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{1-[\xi
r_{n}],r_{n}-[\xi
r_{n}]}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]\;,\ \
\xi\in(0,1)\;.$

Note that the third equality follows by stationarity. By (5.4), for each
$\xi\in(0,1)$, $h_{n,0}(\xi)\to{\boldsymbol{\nu}}^{*}(H\widetilde{H})$.
Furthermore, the sequence $\\{h_{n,0},n\geq 1\\}$ is uniformly bounded in $n$
and $\xi$. Thus, with help of 1,

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{0})=\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}[H_{n,0}^{\mathcal{A}}\widetilde{H}_{0}]={\boldsymbol{\nu}}^{*}(H\widetilde{H})\;.$

The other covariances vanish. Indeed, we analyse
$\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{j})$, $j\geq 1$. We have,
using again the stationarity as above,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}[H_{n,0}^{\mathcal{A}},\widetilde{H}_{j}]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{jr_{n}+1,(j+1)r_{n}}/u_{n}\right)\right]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=1}^{r_{n}}\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{jr_{n}+1-i,(j+1)r_{n}-i}/u_{n}\right)\right]$
$\displaystyle=\int_{0}^{1}h_{n,j}(\xi)\mathrm{d}\xi$

with a function $h_{n,j}$ defined on $(0,1)$ by

$\displaystyle
h_{n,j}(\xi)=\mathbb{E}\left[H^{\mathcal{A}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\widetilde{H}\left(\boldsymbol{X}_{jr_{n}-[\xi
r_{n}]+1,(j+1)r_{n}-[\xi
r_{n}]}/u_{n}\right)\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]\;.$

Until now we proceeded as in the case $j=0$ above. However, now we use (5.2).
For each $\xi\in(0,1)$, $jr_{n}-[\xi r_{n}]\to+\infty$. Hence,
$h_{n,j}(\xi)\to 0$. Bounded convergence and 1 give

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{j})=0\;.$
(5.21)

The same idea applies to
$\mathrm{cov}(\widetilde{H}_{0},H_{n,j}^{\mathcal{A}})$, $j\geq 1$:

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}(\widetilde{H}_{0},H_{n,j}^{\mathcal{A}})=0\;.$
(5.22)

Now, by (5.21)-(5.22), the terms that correspond to $j=1,2$ in (5.3) vanish,
while (5.19) and (3.6c) give

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=3}^{m_{n}-1}\left\\{\mathrm{cov}(H_{n,0}^{\mathcal{A}},\widetilde{H}_{j})+\mathrm{cov}(\widetilde{H}_{0},H_{n,j}^{\mathcal{A}})\right\\}$
$\displaystyle=O(1)\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{\infty}\beta_{jr_{n}}=o(1)\;.$

∎

###  5.4 Empirical cluster process of runs statistics

Recall that

$\displaystyle
H^{\mathcal{A}}(\boldsymbol{x})=H(\boldsymbol{x})\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x})=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>1\right\\}}\;.$

Define

$\displaystyle
H_{s}^{\mathcal{A}}(\boldsymbol{x})=H(\boldsymbol{x}/s)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x}/s)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}}\;.$
(5.23)

Recall that $0<s_{0}<1<t_{0}<\infty$. Recall also that
$k_{n}=n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$. Define also the
classical tail empirical process by

$\displaystyle\mathbb{T}_{n}(s)=\sqrt{k_{n}}\left\\{\frac{\sum_{j=1}^{n}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{j}\right|>su_{n}\right\\}}}{k_{n}}-s^{-\alpha}\right\\}\;,\
\ s\in[s_{0},t_{0}]\;.$

In order to deal with asymptotic normality of runs estimators, we study the
empirical process

$\displaystyle\mathbb{F}_{n}(H_{s}^{\mathcal{A}})$
$\displaystyle:=\sqrt{k_{n}}\left\\{{\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H_{s}^{\mathcal{A}})-{\boldsymbol{\nu}}^{*}(H_{s})\right\\}$
$\displaystyle=\sqrt{k_{n}}\left\\{\frac{\sum_{i=1}^{n}H_{s}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)}{k_{n}}-s^{-\alpha}{\boldsymbol{\nu}}^{*}(H)\right\\}\;.$

The process $\mathbb{F}_{n}(H_{s}^{\mathcal{A}})$ is viewed as a random
element with values in $\mathbb{D}([s_{0},t_{0}])$. The next result is crucial
to establish convergence of runs estimators.

######  Theorem 5.5.

Let $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ be a stationary, regularly
varying $\mathbb{R}^{d}$-valued time series. Assume that 1,
$\beta^{\prime}(r_{n})$, $\mathcal{S}(r_{n},u_{n})$, (3.11) and (3.12b) hold.
Suppose that Assumption 3.4 is satisfied.

Then ${\mathbb{F}}_{n}(H^{\mathcal{A}}_{\cdot})$ converges weakly in
$(\mathbb{D}([s_{0},t_{0}]),J_{1})$ to a Gaussian process
${\mathbb{G}}(H_{\cdot})$ with the covariance
${\boldsymbol{\nu}}^{*}(H_{s}H_{t})$. If moreover ${\rm ANSJB}(r_{n},u_{n})$
is satisfied, then the convergence holds for $H\in\mathcal{B}$. If
additionally (3.12a) is satisfied, then the processes
${\mathbb{F}}_{n}(H^{\mathcal{A}}_{\cdot})$ and ${\mathbb{T}}_{n}(\cdot)$
converge jointly
$({\mathbb{G}}(H_{\cdot}),{\mathbb{G}}(\mathcal{E}_{\cdot}))$.

###  5.5 Proof of Theorem 3.5

Write $\psi_{n}={\left|\boldsymbol{X}\right|}_{(n:n-k_{n})}/u_{n}$. Since
$k_{n}=n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$, we can rewrite
$\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})$ as
$\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})={\widetilde{\boldsymbol{\xi}}}^{*}_{n,r_{n}}(H^{\mathcal{A}}_{\psi_{n}})$
(cf. (3.4)-(3.5)). Therefore,

$\displaystyle\sqrt{k_{n}}\left\\{\widehat{\boldsymbol{\xi}}^{*}_{n,r_{n}}(H^{\mathcal{A}})-{\boldsymbol{\nu}}^{*}(H)\right\\}={\mathbb{F}}_{n}(H^{\mathcal{A}}_{\psi_{n}})+\sqrt{k_{n}}\left\\{{\boldsymbol{\nu}}^{*}(H_{\psi_{n}})-{\boldsymbol{\nu}}^{*}(H)\right\\}\;.$

We have local uniform convergence of
$\\{\mathbb{F}_{n}(H^{\mathcal{A}}_{s}),\;s\in[s_{0},\,t_{0}]\\}$ to a
continuous Gaussian process ${\mathbb{G}}$ thanks to Theorem 5.5. Moreover,
the convergence of $\\{{\mathbb{T}}_{n}(\cdot),\;s\in[s_{0},\,t_{0}]\\}$
yields $\psi_{n}\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}1$, jointly with $\mathbb{F}_{n}(H^{\mathcal{A}}_{s})$.
Therefore,
$\mathbb{F}_{n}(H^{\mathcal{A}}_{\psi_{n}})\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}{\mathbb{G}}(H)$. Using Vervaat’s theorem, we have,
jointly with the previous convergence,
$\sqrt{k_{n}}(\psi_{n}^{-\alpha}-1)\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}-{\mathbb{G}}(\mathcal{E})$. Therefore, by the
homogeneity of ${\boldsymbol{\nu}}^{*}$,

$\displaystyle\sqrt{k_{n}}\left\\{{\boldsymbol{\nu}}^{*}(H_{\psi_{n}})-{\boldsymbol{\nu}}^{*}(H)\right\\}={\boldsymbol{\nu}}^{*}(H)\sqrt{k_{n}}(\psi_{n}^{-\alpha}-1)\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}-{\boldsymbol{\nu}}^{*}(H){\mathbb{G}}(\mathcal{E}).$

Since the convergence hold jointly, we conclude the result.

###  5.6 Proof of Theorem 5.5 \- fidi convergence

Recall the disjoint blocks of size $r_{n}$ (cf. (3.2)):

$\displaystyle J_{j}:=\\{jr_{n}+1,\ldots,(j+1)r_{n}\\}\;,\ \
j=0,\ldots,m_{n}-1\;.$

These blocks were chosen to calculate the limiting covariance of the process
$\mathbb{F}_{n}$. However, they are not appropriate for a proof of the central
limit theorem. We need to introduce a large-small blocks decomposition.

For this purpose let $z_{n}$ be a sequence of integers such that
$z_{n}\to\infty$ and

$\displaystyle\lim_{n\to\infty}z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})=\lim_{n\to\infty}\frac{z_{n}r_{n}}{\sqrt{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}}=\lim_{n\to\infty}\frac{z_{n}r_{n}}{\sqrt{k_{n}}}=0\;.$
(5.24)

This is possible thanks to the assumptions 1 and (3.11). We note that this
assumption is needed for the Lindeberg condition only. Set

$\widetilde{m}_{n}=\frac{q_{n}}{(z_{n}+3)r_{n}}=\frac{n-r_{n}}{(z_{n}+3)r_{n}}\sim\frac{n}{z_{n}r_{n}}$

and assume for simplicity that $\widetilde{m}_{n}$ is an integer. Since
$z_{n}\to\infty$, we have $\widetilde{m}_{n}=o(m_{n})$. For
$j=1,\ldots,\widetilde{m}_{n}$ define now large and small blocks as follows:

$\displaystyle L_{1}=\\{1,\ldots,z_{n}r_{n}\\}\;,\ \
S_{1}=\\{z_{n}r_{n}+1\ldots,z_{n}r_{n}+3r_{n}\\}\;,$ $\displaystyle
L_{2}=\\{z_{n}r_{n}+3r_{n}+1,\ldots,2z_{n}r_{n}+3r_{n}\\}\;,\ \
S_{2}=\\{2z_{n}r_{n}+3r_{n}+1,\ldots,2z_{n}r_{n}+6r_{n}\\}\;,$ $\displaystyle
L_{j}=\\{(j-1)z_{n}r_{n}+3(j-1)r_{n}+1,\ldots,jz_{n}r_{n}+3(j-1)r_{n}\\}\;,$
$\displaystyle
S_{j}=\\{jz_{n}r_{n}+3(j-1)r_{n}+1,\ldots,jz_{n}r_{n}+3jr_{n}\\}\;.$

The block $L_{1}$ is obtained by merging $z_{n}$ consecutive blocks
$J_{0},\ldots,J_{z_{n}-1}$ of size $r_{n}$. Likewise, $S_{1}=J_{z_{n}}\cup
J_{z_{n}+1}\cup J_{z_{n}+2}$. Therefore, the large block of size $z_{n}r_{n}$
is followed by the small block of size $3r_{n}$, which in turn is followed by
the large block of size $z_{n}r_{n}$ and so on. All together,

$\displaystyle\bigcup_{j=1}^{\widetilde{m}_{n}}\left(L_{j}\cup
S_{j}\right)=\\{1,\ldots,q_{n}\\}=\\{1,\ldots,n-r_{n}\\}\;.$

Write

$\displaystyle\sum_{i=1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=\sum_{i=1}^{q_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)+\sum_{i=q_{n}+1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)$
$\displaystyle=\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(l)}(H^{\mathcal{A}})+\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(s)}(H^{\mathcal{A}})+W_{n}\;,$
(5.25)

where now

$\displaystyle\Psi_{j}^{(l)}(H^{\mathcal{A}})=\sum_{i\in
L_{j}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;,\ \
\Psi_{j}^{(s)}(H^{\mathcal{A}})=\sum_{i\in
S_{j}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;$

and

$W_{n}=\sum_{i=q_{n}+1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=\sum_{i=n-r_{n}+1}^{n}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;.$

With such the decomposition,
$\boldsymbol{X}_{1-r_{n}},\ldots,\boldsymbol{X}_{z_{n}r_{n}+r_{n}}$ used in
the definition of $\Psi_{1}^{(l)}(H^{\mathcal{A}})$ are separated by at least
$r_{n}$ from the random variables that define
$\Psi_{2}^{(l)}(H^{\mathcal{A}})$. The mixing condition (3.6a) allows us to
replace $\boldsymbol{X}$ with the independent blocks process, that is, we can
treat the random variables $\Psi_{j}^{(l)}(H^{\mathcal{A}})$,
$j=1,\ldots,\widetilde{m}_{n}$, as independent. The same applies to
$\Psi_{j}^{(s)}(H^{\mathcal{A}})$.

Set

$\displaystyle{\mathbb{Z}}_{n}(H^{\mathcal{A}})=\sum_{j=1}^{\widetilde{m}_{n}}\left\\{Z_{n,j}(H^{\mathcal{A}})-\mathbb{E}[Z_{n,j}(H^{\mathcal{A}})]\right\\}=:\sum_{j=1}^{\widetilde{m}_{n}}\bar{Z}_{n,j}(H^{\mathcal{A}})$
(5.26)

with

$\displaystyle
Z_{n,j}(H^{\mathcal{A}})=\frac{1}{\sqrt{k_{n}}}\Psi_{j}^{(l)}(H^{\mathcal{A}})\;.$
(5.27)

The next steps are standard.

  * •

First, we show that the limiting variance of the large blocks process
${\mathbb{Z}_{n}}$ is the same as that of the process $\mathbb{F}_{n}$;

  * •

Next, we show that the small blocks process (the scaled second term in (5.6))
is negligible;

  * •

We show that the boundary term $W_{n}$ is also negligible;

  * •

Finally, we will verify the Lindeberg condition for the large blocks process.

Variance of the large blocks. We have (using the assumed independence of
$\Psi_{j}^{(l)}(H^{\mathcal{A}})$)

$\displaystyle\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(l)}(H^{\mathcal{A}})\right)=\frac{\widetilde{m}_{n}}{k_{n}}\mathrm{var}(\Psi_{1}^{(l)}(H^{\mathcal{A}}))$
$\displaystyle\sim\frac{1}{z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{var}\left(\sum_{i=1}^{z_{n}r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\right)$
$\displaystyle=\frac{1}{z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{var}\left(\sum_{j=0}^{z_{n}-1}H_{n,j}^{\mathcal{A}}\right)\;,$
(5.28)

where $H_{n,j}^{\mathcal{A}}$ is defined in (3.3) and where in the last line
we decomposed the block $L_{1}=\\{1,\ldots,z_{n}r_{n}\\}$ into $z_{n}$
disjoint blocks $J_{0},\ldots,J_{z_{n}-1}$ and $\widetilde{m}_{n}\sim
m_{n}/z_{n}$. The next steps follow easily from (5.3) with $m_{n}$ replaced by
$z_{n}$.

The term in (5.6) becomes

$\displaystyle\frac{\mathrm{var}\left(H_{n,0}^{\mathcal{A}}\right)}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}+\frac{2}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{z_{n}-1}\left(1-\frac{j}{z_{n}}\right)\mathrm{cov}(H_{n,0}^{\mathcal{A}},H_{n,j}^{\mathcal{A}})\;.$
(5.29)

It follows immediately from (5.3) that the limit of the first term above is

$\displaystyle\lim_{n\to\infty}\frac{\mathrm{var}\left(H_{n,0}^{\mathcal{A}}\right)}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}={\boldsymbol{\nu}}^{*}(H^{2})\;.$
(5.30)

Now, for the second term in (5.29) we adapt the proof of Lemma 3.2 from
$m_{n}$ to $z_{n}$.

As in (5.3), for $j\geq 1$,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\left|\mathrm{cov}(H_{n,0}^{\mathcal{A}},{H}_{n,j}^{{\mathcal{A}}})\right|$
$\displaystyle\leq\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}|{g}_{n}(h)|=:I_{j}$

with (this time)

$\displaystyle{g}_{n}(h)$
$\displaystyle=\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}(H^{\mathcal{A}}(\boldsymbol{X}_{-r_{n},r_{n}}),{H}^{\mathcal{A}}(\boldsymbol{X}_{h-r_{n},h+r_{n}}))\;.$

For $h>2r_{n}$, similarly to (5.18), we have by (5.1),

$\displaystyle|{g}_{n}(h)|$
$\displaystyle\leq\frac{\|H\|_{\infty}^{2}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\beta_{h-2r_{n}}\;.$

Thus,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{z_{n}-1}|\mathrm{cov}(H_{n,0}^{\mathcal{A}},{H}_{n,j}^{\mathcal{A}})|\leq\frac{\|H\|_{\infty}^{2}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{z_{n}-1}\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}\beta_{h-2r_{n}}$
$\displaystyle\leq
2\frac{\|H\|_{\infty}^{2}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=3r_{n}+1}^{\infty}\beta_{h-2r_{n}}=O(1)\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=r_{n}+1}^{\infty}\beta_{i}=o(1)$

by the assumption (3.6b). The terms that correspond to $j=1,2,3$ in (5.3) are
negligible.

In summary, we showed that

$\lim_{n\to\infty}\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(l)}(H^{\mathcal{A}})\right)={\boldsymbol{\nu}}^{*}(H^{2})\;.$

Variance of the small blocks. We have (using again the assumed independence of
$\Psi_{j}^{(s)}(H^{\mathcal{A}})$ thanks to the beta-mixing)

$\displaystyle\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(s)}(H^{\mathcal{A}})\right)=\frac{\widetilde{m}_{n}}{k_{n}}\mathrm{var}(\Psi_{1}^{(s)}(H^{\mathcal{A}}))\sim\frac{1}{z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{var}(\Psi_{1}^{(s)}(H^{\mathcal{A}}))\;.$

Since the size of $\Psi_{1}^{(s)}(H^{\mathcal{A}})$ is $3$ times the size of
$H_{n,1}^{\mathcal{A}}$ defined in (3.3), we have by (5.3)

$\displaystyle\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}\sum_{j=1}^{\widetilde{m}_{n}}\Psi_{j}^{(s)}(H^{\mathcal{A}})\right)\sim\frac{1}{z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}){\boldsymbol{\nu}}^{*}(H^{2})=O(1/z_{n})=o(1)\;.$

Variance of the boundary term $W_{n}$. We have (cf. (3.3))

$\displaystyle\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}W_{n}\right)=\mathrm{var}\left(\frac{1}{\sqrt{k_{n}}}\sum_{i=1}^{r_{n}}H^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\right)=\frac{\mathrm{var}(H_{n,0}^{\mathcal{A}})}{k_{n}}=\frac{\mathrm{var}(H_{n,0}^{\mathcal{A}})}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;.$

The latter term vanishes when $n\to\infty$, using (5.30) and $r_{n}/n\to 0$.

Lindeberg condition for ${\mathbb{Z}}_{n}(H^{\mathcal{A}})$. We need to show
that for all $\eta>0$,

$\displaystyle\lim_{n\to\infty}\widetilde{m}_{n}\mathbb{E}\left[Z_{n,1}^{2}(H^{\mathcal{A}})\mathbbm{1}{\left\\{|Z_{n,1}(H^{\mathcal{A}})|>\eta\right\\}}\right]=0\;.$
(5.31)

Since $H$ is bounded, then by (5.24),

$\displaystyle|Z_{n,1}(H^{\mathcal{A}})|\leq\frac{\sqrt{k_{n}}z_{n}r_{n}}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\|H\|_{\infty}\sim\frac{z_{n}r_{n}}{\sqrt{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}}\|H\|_{\infty}=o(1)\;.$
(5.32)

Thus, the indicator in (5.31) becomes zero for large $n$.

###  5.7 Proof of Theorem 5.5 \- asymptotic equicontinuity

We need the following lemma which is an adapted version of Theorem 2.11.1 in
[vdVW96]. Let $\mathbb{Z}_{n}$ be the empirical process indexed by a semi-
metric space $(\mathcal{G},\rho)$, defined by

$\displaystyle\mathbb{Z}_{n}(f)=\sum_{j=1}^{\widetilde{m}_{n}}\left\\{Z_{n,j}(f)-\mathbb{E}[Z_{n,j}(f)]\right\\}\;,$

where $\\{Z_{n,j},n\geq 1\\}$, $j=1,\ldots,\widetilde{m}_{n}$, are i.i.d.
separable, stochastic processes and $\widetilde{m}_{n}$ is a sequence of
integers such that $\widetilde{m}_{n}\to\infty$. Define the random semi-metric
$d_{n}$ on $\mathcal{G}$ by

$\displaystyle
d_{n}^{2}(f,g)=\sum_{j=1}^{\widetilde{m}_{n}}\\{Z_{n,j}(f)-Z_{n,j}(g)\\}^{2}\;,f,g\in\mathcal{G}\;.$

######  Lemma 5.6.

Assume that $(\mathcal{G},\rho)$ is totally bounded. Assume moreover that:

  1. (i)

For all $\eta>0$,

$\displaystyle\lim_{n\to\infty}{\widetilde{m}_{n}}\mathbb{E}[\|Z_{n,1}\|^{2}_{\mathcal{G}}\mathbbm{1}{\left\\{\|Z_{n,1}\|^{2}_{\mathcal{G}}>\eta\right\\}}]=0\;.$
(5.33)

  2. (ii)

For every sequence $\\{\delta_{n}\\}$ which decreases to zero,

$\displaystyle\lim_{n\to\infty}\sup_{f,g\in\mathcal{G}\atop\rho(f,g)\leq\delta_{n}}\mathbb{E}[d_{n}^{2}(f,g)]=0\;.$
(5.34)

  3. (iii)

There exists a measurable majorant $N^{*}(\mathcal{G},d_{n},\epsilon)$ of the
covering number $N(\mathcal{G},d_{n},\epsilon)$ such that for every sequence
$\\{\delta_{n}\\}$ which decreases to zero,

$\displaystyle\int_{0}^{\delta_{n}}\sqrt{\log
N^{*}(\mathcal{G},d_{n},\epsilon)}\mathrm{d}\epsilon\stackrel{{\scriptstyle\tiny\mathbb{P}}}{{\longrightarrow}}0\;.$

Then $\\{\mathbb{Z}_{n},n\geq 1\\}$ is asymptotically $\rho$-equicontinuous,
i.e. for each $\eta>0$,

$\displaystyle\lim_{\delta\to
0}\limsup_{n\to\infty}\mathbb{P}\left(\sup_{{f,g\in\mathcal{G}}\atop{\rho(f,g)<\delta}}|\mathbb{Z}_{n}(f)-\mathbb{Z}_{n}(g)|>\eta\right)=0\;.$

######  Remark 5.7.

The separability assumption is not in [vdVW96]. It implies measurability of
$\|Z_{n,1}\|_{\mathcal{G}}$. Furthermore, the separability also implies that
for all $\delta>0$, $n\in\mathbb{N}$, $(e_{j})_{1\leq
j\leq\widetilde{m}_{n}}\in\\{-1,0,1\\}^{\widetilde{m}_{n}}$ and
$i\in\\{1,2\\}$, the supremum

$\displaystyle\sup_{f,g\in\mathcal{G}\atop\rho(f,g)<\delta}\left|\sum_{j=1}^{\widetilde{m}_{n}}e_{j}\left(Z_{n,j}(f)-Z_{n,j}(g)\right)^{i}\right|$
$\displaystyle=\sup_{f,g\in\mathcal{G}_{0}\atop\rho(f,g)<\delta}\left|\sum_{j=1}^{\widetilde{m}_{n}}e_{j}\left(Z_{n,j}(f)-Z_{n,j}(g)\right)^{i}\right|\;$

is measurable, which is an assumption of [vdVW96]. $\oplus$

####  5.7.1 Asymptotic equicontinuity of the empirical process of sliding
blocks

Recall the big-blocks process ${\mathbb{Z}}_{n}(H^{\mathcal{A}})$ (cf.
(5.26)-(5.27)). Recall also that thanks to the $\beta$-mixing we can consider
random variables $\Psi_{j}^{(l)}(H^{\mathcal{A}})$,
$j=1,\ldots,\widetilde{m}_{n}$ to be independent. Recall that
$H^{\mathcal{A}}_{s}$ is defined in (5.23). We need to prove the asymptotic
equicontinuity of ${\mathbb{Z}}_{n}(H_{s}^{\mathcal{A}})$ indexed by the class
$\mathcal{G}=\\{H^{\mathcal{A}}_{s},s\in[s_{0},t_{0}]\\}$ equipped with the
metric
$\boldsymbol{\rho}^{*}(H_{s}^{\mathcal{A}},H_{t}^{\mathcal{A}})={\boldsymbol{\nu}}^{*}(\\{H_{s}^{\mathcal{A}}-{H}_{t}^{\mathcal{A}}\\}^{2})$.
The same argument can be used to prove the asymptotic equicontinuity for the
small blocks process. This yields asymptotic equicontinuity of
${\mathbb{F}}_{n}(H^{\mathcal{A}}_{s})$.

In what follows, the proof of the Lindeberg-type condition (5.33) is easy. The
proof of (5.34) is quite involved.

Thanks to Assumption 3.4, the condition (3.10) is satisfied. Its validity is
discussed in Section 5.8.

##### Lindeberg condition: Proof of (5.33).

We re-write (5.32) as follows:

$\displaystyle\sup_{s\in[s_{0},t_{0}]}|Z_{n,1}(H_{s}^{\mathcal{A}})|\leq\frac{\sqrt{k_{n}}z_{n}r_{n}}{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\|H\|_{\infty}\leq\frac{z_{n}r_{n}}{\sqrt{n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}}\sup_{s\in[s_{0},t_{0}]}\|H_{s}\|_{\infty}\;,$

Since the class $\\{H_{s}:s\in[s_{0},t_{0}]\\}$ is linearly ordered,
$\sup_{s\in[s_{0},t_{0}]}\|H_{s}\|_{\infty}$ is achieved either at $s=s_{0}$
or $s=t_{0}$. Hence, the Lindeberg condition i of Lemma 5.6 holds by (5.31).

##### Asymptotic continuity of random semi-metric: Proof of (5.34).

The proof is rather long and technical.

Define the random metric

$\displaystyle
d_{n}^{2}(H_{s}^{\mathcal{A}},{H}_{t}^{\mathcal{A}})=\sum_{j=1}^{\widetilde{m}_{n}}(Z_{n,j}(H_{s}^{\mathcal{A}})-Z_{n,j}({H}_{t}^{\mathcal{A}}))^{2}\;.$

Let (cf. (3.3))

$\displaystyle H_{s,n,j}^{\mathcal{A}}$
$\displaystyle=\sum_{i=jr_{n}+1}^{(j+1)r_{n}}H\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)=i\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{i}\right|>su_{n}\right\\}}\;.$

We need to evaluate
$\mathbb{E}[d_{n}^{2}(H^{\mathcal{A}}_{s},H^{\mathcal{A}}_{t})]$:

$\displaystyle\mathbb{E}[d_{n}^{2}(H^{\mathcal{A}}_{s},H^{\mathcal{A}}_{t})]$
$\displaystyle=\frac{k_{n}\widetilde{m}_{n}}{(n\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}))^{2}}\mathbb{E}\left[\left(\sum_{i=1}^{z_{n}r_{n}}\left\\{H^{\mathcal{A}}_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-H^{\mathcal{A}}_{t}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\right\\}\right)^{2}\right]$
$\displaystyle\sim\frac{1}{z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}\left[\left(\sum_{j=0}^{z_{n}-1}\left\\{H_{s,n,j}^{\mathcal{A}}-H_{t,n,j}^{\mathcal{A}}\right\\}\right)^{2}\right]\;,$
(5.35)

where in the last line we decomposed the block $L_{1}$ into $z_{n}$ disjoint
blocks $J_{0},\ldots,J_{z_{n}-1}$, $\widetilde{m}_{n}\sim m_{n}/z_{n}$; cf.
(5.6). The term in (5.7.1) becomes

$\displaystyle\frac{\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}^{2}]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
$\displaystyle+2\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{z_{n}-1}\left(1-\frac{j}{z_{n}}\right)\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}\left\\{H_{s,n,j}^{\mathcal{A}}-H_{t,n,j}^{\mathcal{A}}\right\\}]$

The above lines correspond to (5.3) with $m_{n}$ replaced by $z_{n}$.

We are going to prove two statements:

$\displaystyle\lim_{n\to\infty}\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\frac{\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}^{2}]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}=0\;$
(5.36)

and

$\displaystyle\lim_{n\to\infty}\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=1}^{z_{n}-1}\left(1-\frac{j}{z_{n}}\right)\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}\left\\{H_{s,n,j}^{\mathcal{A}}-H_{t,n,j}^{\mathcal{A}}\right\\}]=0\;.$
(5.37)

##### Proof of (5.36).

We will write $\\{H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}(\boldsymbol{x})$
for $H_{s}^{\mathcal{A}}(\boldsymbol{x})-H_{t}^{\mathcal{A}}(\boldsymbol{x})$.

Similarly to (5.3),

$\displaystyle\frac{\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}^{2}]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
$\displaystyle\leq\|H\|\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=-r_{n}}^{r_{n}}\left|\mathbb{E}[\\{H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\times\\{{H}_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)]\right|$
$\displaystyle=:\|H\|\sum_{h=-r_{n}}^{r_{n}}|g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|\;$
(5.38)

with

$\displaystyle|{g}_{n}(h,G)|=\left|\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}[G(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})G(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})]\right|\;.$
(5.39)

Using the definition (5.23) of $H_{s}^{\mathcal{A}}$, the fact that $s,t\geq
s_{0}$ and since $H$ is bounded, we immediately get

$\displaystyle|{g}_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|\leq\frac{4}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\|H\|^{2}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n},\left|\boldsymbol{X}_{h}\right|>s_{0}u_{n})\;.$
(5.40)

To get a more precise bound that involves the difference $s-t$ we need to
consider three cases. The reason for this is that we need to keep the absolute
value in (5.39) outside of the expectation. As such, computations below are
quite technically involved.

To shorten our displays, we introduce the notation

$\displaystyle\mathcal{I}(i,s):=\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{i}\right|>su_{n}\right\\}}\;.$
(5.41)

##### Case 1.

Assume here that $\mathcal{A}$ is $0$-homogeneous. Then for any $i$,

$\displaystyle
H_{s}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)$
$\displaystyle=H\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)=i\right\\}}\mathcal{I}(i,s)$
(5.42)
$\displaystyle=H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=i\right\\}}\mathcal{I}(i,s)\;;$

(we keep $u_{n}$ in the argument of $\mathcal{A}$, although it can be
omitted). What is important in this decomposition is that we can control
monotonicity (withe respect to $s$) of each term.

Then

$\displaystyle
H_{s}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-H_{t}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=$
$\displaystyle=\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=i\right\\}}H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\Big{(}\mathcal{I}(i,s)-\mathcal{I}(i,t)\Big{)}$
$\displaystyle\phantom{=}+\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=i\right\\}}\mathcal{I}(i,t)\Big{(}H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-H_{t}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\Big{)}$
$\displaystyle=:T_{1}(i)+T_{2}(i)$

and hence

$\displaystyle\left|\mathbb{E}\left[\\{H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\\{H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)\right]\right|$

is bounded by the sum of four nonnegative terms $W_{11}+W_{22}+W_{12}+W_{21}$
that we are going to define below. The general idea is that we will obtain
rough bounds in terms of the difference
$\Big{(}\mathcal{I}(i,s)-\mathcal{I}(i,t)\Big{)}$, except of one case which
involves the product $T_{2}(0)T_{2}(h)$. Coming back to the definitions of
$W_{i,j}$, the double sub-index $\\{12\\}$ of $W$ indicates that $W_{12}$ is
related to multiplying $T_{1}(0)$ by $T_{2}(h)$; $W_{21}$ means we multiply
$T_{2}(0)$ and $T_{1}(h)$):

$\displaystyle
W_{11}:=\|H\|^{2}\mathbb{E}\left[\left(\mathcal{I}(0,s)-\mathcal{I}(0,t)\right)\left(\mathcal{I}(h,s)-\mathcal{I}(h,t)\right)\right]\;,$

(above, the indicators of the anchoring map are omitted);

$\displaystyle W_{22}:=$
$\displaystyle\mathbb{E}\left[\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)=0\right\\}}\mathcal{I}(0,t)\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)\right]\;,$

(above,
$\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)=h\right\\}}$
and $\mathcal{I}(h,t)$ are omitted);

$\displaystyle
W_{12}:=\pm\|H\|\mathbb{E}\left[\Big{(}\mathcal{I}(0,s)-\mathcal{I}(0,t)\Big{)}\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)\right]\;,$

(both indicators of the anchoring maps and $\mathcal{I}(h,t)$ are omitted);

$\displaystyle
W_{21}:=\pm\|H\|\mathbb{E}\left[\Big{(}\mathcal{I}(h,s)-\mathcal{I}(h,t)\Big{)}\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\right]\;,$

(both indicators of the anchoring maps and $\mathcal{I}(0,t)$ are omitted).

Note that the right-hand side of both $W_{11},W_{22}$ is nonnegative thanks to
the monotonicity and $s<t$, while for the right-hand side of $W_{12},W_{21}$
we need to put $\pm$, since the sign of the expressions there depends on
whether the map $s\to H_{s}$ is decreasing or increasing.

Thus,

$\displaystyle
W_{11}\leq\|H\|^{2}\left(\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}s)-\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}t)\right)\;,$
(5.43) $\displaystyle W_{22}\leq\pm
2\|H\|\mathbb{E}\left[\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)=0\right\\}}\mathcal{I}(0,t)\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\right]\;,$
(5.44) $\displaystyle W_{12}+W_{21}\leq
4\|H\|^{2}\left(\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}s)-\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}t)\right)\;.$
(5.45)

The bound on $W_{22}$ (with $+$) is obvious if $H_{s}-H_{t}\geq 0$ (thus,
$s\to H_{s}$ is decreasing), while in an increasing case of $s\to H_{s}$ we
use the following observation: if $a,b<0$, $|b|<c$, then $ab\leq-ac$ (yielding
$-$ on the right-hand side of (5.44)). The bound on $W_{12}+W_{21}$ follows
from the same reasoning.

In summary, with $g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})$ defined in
(5.39), we have

$\displaystyle|g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|\leq
5\|H\|^{2}\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}s)-\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n}t)}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
(5.46) $\displaystyle\pm
2\|H\|\frac{\mathbb{E}\left[\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n}\right\\}}\left\\{H_{s}-H_{t}\right\\}\left(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}\right)\right]}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;,$
(5.47)

where again the presence of $\pm$ depends on the sign of $H_{s}-H_{t}$.

We can ignore the scaling factor $2\|H\|$ in (5.47) and write it as (recall
that the anchoring map is $0$-homogeneous)

$\displaystyle\mathbb{E}[H_{s/s_{0}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)=0\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n}]\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n})}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
$\displaystyle\phantom{=}-\mathbb{E}[H_{t/s_{0}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)=0\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n}]\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n})}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
$\displaystyle=\Big{(}\mu_{n,r_{n}}(s)-\mu_{n,r_{n}}(t)\Big{)}\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n})}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}=:\Big{(}\tilde{\mu}_{n,r_{n}}(s)-\tilde{\mu}_{n,r_{n}}(t)\Big{)}\;,$

with

$\displaystyle\mu_{n,r_{n}}(\cdot)$
$\displaystyle=\mathbb{E}[H_{\cdot/s_{0}}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}\left(\boldsymbol{X}_{-r_{n},r_{n}}/(s_{0}u_{n})\right)=0\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n}]\;$

and

$\tilde{\mu}_{n,r_{n}}(s)=\mu_{n,r_{n}}(s)\frac{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n})}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\;.$

Thanks to (5.5),
$\lim_{n\to\infty}\mu_{n,r_{n}}(s)={\boldsymbol{\nu}}^{*}(H_{s/s_{0}})$.
Thanks to the monotonicity of $s\to H_{S}$ and homogeneity of
${\boldsymbol{\nu}}^{*}$, the convergence of $\tilde{\mu}_{n,r_{n}}(s)$ to

$s_{0}^{-\alpha}{\boldsymbol{\nu}}^{*}(H_{s/s_{0}})=s_{0}^{-\alpha}(s/s_{0})^{-\alpha}{\boldsymbol{\nu}}^{*}(H)=s^{-\alpha}{\boldsymbol{\nu}}^{*}(H)$

is uniform on $[s_{0},t_{0}]$. Thus, for $s,t\in[s_{0},t_{0}]$,

$\left|\tilde{\mu}_{n,r_{n}}(s)-\tilde{\mu}_{n,r_{n}}(t)\right|\leq
2\sup_{s_{0}\leq u\leq
t_{0}}\left|\tilde{\mu}_{n,r_{n}}(u)-{\boldsymbol{\nu}}^{*}(H_{u})\right|+{\boldsymbol{\nu}}^{*}(H)\\{s^{-\alpha}-t^{-\alpha}\\}\;.$

Fix $\eta>0$. For large enough $n$, the uniform convergence yields

$\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\left|\tilde{\mu}_{n,r_{n}}(s)-\tilde{\mu}_{n,r_{n}}(t)\right|\leq\eta+{\boldsymbol{\nu}}^{*}(H)\sup_{s_{0}\leq
s,t\leq t_{0}\atop|s-t|\leq\delta_{n}}\\{s^{-\alpha}-t^{-\alpha}\\}\\\
\leq\eta+\alpha s_{0}^{-\alpha-1}\delta_{n}{\boldsymbol{\nu}}^{*}(H)\;.$
(5.48)

The uniform convergence also yields that the term in (5.46) is bounded by
$\eta+\alpha s_{0}^{-\alpha-1}\delta_{n}$. This, together with (5.48), gives

$\displaystyle|g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|\leq\eta+\mathrm{cst}\
\delta_{n}$ (5.49)

with a generic constant $\mathrm{cst}$.

Fix an integer $r$. Using (5.40) and (5.49) we have

$\displaystyle\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\frac{\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}^{2}]}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}$
$\displaystyle\leq\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\sum_{h=-r}^{r}|g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|+\sup_{s_{0}\leq
s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}\sum_{|h|=r}^{r_{n}}|g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})|$
$\displaystyle\leq\mathrm{cst}\
r(\eta+\delta_{n})+\frac{4}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{|h|=r}^{r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n},\left|\boldsymbol{X}_{h}\right|>s_{0}u_{n})\;.$

Applying the anticlustering conditions $\mathcal{S}(r_{n},u_{n})$ to the
second term, letting $\delta_{n}\to 0$, since $\eta$ is arbitrary, this proves
(5.36).

##### Case 2.

Now, we consider the anchoring maps $\mathcal{A}^{(1)}$ and
$\mathcal{A}^{(2)}$ which are not 0-homogeneous. Note that for $j=1,2$ we can
write

$\displaystyle
H_{s}^{\mathcal{A}^{(j)}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)$
$\displaystyle=H\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)\mathbbm{1}{\left\\{\mathcal{A}^{(j)}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/(su_{n})\right)=i\right\\}}\mathcal{I}(i,s)$
$\displaystyle=H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\mathcal{I}(i,s)F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\;,$

where

$\displaystyle
F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=\mathbbm{1}{\left\\{\boldsymbol{X}_{i-r_{n},i-1}^{*}\leq
su_{n}\right\\}}$

or

$\displaystyle
F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=\mathbbm{1}{\left\\{\boldsymbol{X}_{i+1,i+r_{n}}^{*}\leq
su_{n}\right\\}}$

in case $j=1$ and $j=2$, respectively. Note that regardless of the
monotonicity of the map $s\to\mathcal{A}_{s}^{(j)}$, the map $s\to F_{s}$ is
always non-decreasing. Then (5.42) gives, for any $i\in\mathbb{N}$, and
$\mathcal{A}=\mathcal{A}^{(1)},\mathcal{A}^{(2)}$,

$\displaystyle
H_{s}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-H_{t}^{\mathcal{A}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)=$
$\displaystyle=H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\bigg{(}\mathcal{I}(i,s)-\mathcal{I}(i,t)\bigg{)}$
$\displaystyle\phantom{=}+\mathcal{I}(i,t)F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\big{(}H_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-H_{t}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\big{)}$
$\displaystyle\phantom{=}+\mathcal{I}(i,t)H_{t}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\bigg{(}F_{s}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)-F_{t}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\bigg{)}$
$\displaystyle=:T_{1}(i)+T_{2}(i)+T_{3}(i)\;.$

Again, in this decomposition we can control monotonicity of each term. The
argument now is very similar to that of Case 1. Hence, it is omitted.

Therefore, (5.36) is proved for $\mathcal{A}^{(j)}$, $j=0,1,2$.

##### Proof of (5.37).

We proceed similarly. Recall that for $j\geq 1$ (cf. (5.3)),

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}[\left\\{H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right\\}\left\\{H_{s,n,j}^{\mathcal{A}}-H_{t,n,j}^{\mathcal{A}}\right\\}]$
$\displaystyle=\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}\left[\sum_{h=1}^{r_{n}}\Big{\\{}H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\Big{\\}}\left(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n}\right)\times\sum_{i=jr_{n}+1}^{(j+1)r_{n}}\Big{\\{}H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\Big{\\}}\left(\boldsymbol{X}_{i-r_{n},i+r_{n}}/u_{n}\right)\right]$
$\displaystyle=\sum_{h=(j-1)r_{n}+1}^{jr_{n}}\left(\frac{h}{r_{n}}-(j-1)\right){g}_{n}(h,H^{\mathcal{A}}_{s}-H^{\mathcal{A}}_{t})+\sum_{h=jr_{n}+1}^{(j+1)r_{n}}\left((j+1)-\frac{h}{r_{n}}\right){g}_{n}(h,H^{\mathcal{A}}_{s}-H^{\mathcal{A}}_{t})$
$\displaystyle\leq\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}{g}_{n}(h,H^{\mathcal{A}}_{s}-H^{\mathcal{A}}_{t})=:I_{j}(s,t)$
(5.50)

with the same $g_{n}$ as in (5.39).

Write ${g}_{n}(h,G)$ as

$\displaystyle\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathrm{cov}[G(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n}),G(\boldsymbol{X}_{h-r_{n},h+r_{n}}/u_{n})]+\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}^{2}[G(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})]$
$\displaystyle=:\widetilde{g}_{n}(h,G)+\frac{1}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\mathbb{E}^{2}[G(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})]\;.$

For $h>2r_{n}$ we have by (5.1),

$\displaystyle|\widetilde{g}_{n}(h,H^{\mathcal{A}}_{s}-H^{\mathcal{A}}_{t})|$
$\displaystyle\leq\frac{\mathrm{cst}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\beta_{h-2r_{n}}\;.$
(5.51)

Thus,

$\displaystyle\frac{1}{r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{z_{n}-1}\left(1-\frac{j}{z_{n}}\right)\mathbb{E}[\left(H_{s,n,0}^{\mathcal{A}}-H_{t,n,0}^{\mathcal{A}}\right)\left(H_{s,n,j}^{\mathcal{A}}-H_{t,n,j}^{\mathcal{A}}\right)]$
$\displaystyle\leq\frac{\mathrm{cst}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{j=4}^{z_{n}-1}\sum_{h=(j-1)r_{n}+1}^{(j+1)r_{n}}\beta_{h-2r_{n}}$
$\displaystyle\phantom{\leq}+\frac{\mathrm{cst}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}z_{n}r_{n}\mathbb{E}^{2}[\\{H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}}\\}(\boldsymbol{X}_{-r_{n},r_{n}}/u_{n})]$
$\displaystyle\leq\frac{\mathrm{cst}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{h=3r_{n}+1}^{\infty}\beta_{h-2r_{n}}+\mathrm{cst}z_{n}r_{n}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n})$
$\displaystyle=\frac{\mathrm{cst}}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})}\sum_{i=r_{n}+1}^{\infty}\beta_{i}+o(1)=o(1)$
(5.52)

uniformly in $s,t\in[s_{0},t_{0}]$. In the last line we applied the assumption
(3.6b), and the assumption (5.24).

The terms that correspond to $j=1,2,3$ in (5.7.1) have to be dealt with
separately. We note that
$I_{1}(s,t)=\sum_{h=1}^{r_{n}}g_{n}(h,H_{s}^{\mathcal{A}}-H_{t}^{\mathcal{A}})$
is bounded by the term in (5.38). Hence,

$\displaystyle\lim_{n\to\infty}\sup_{s_{0}\leq s,t\leq
t_{0}\atop|s-t|\leq\delta_{n}}I_{1}(s,t)=0\;.$ (5.53)

Next, using (5.40) and $\mathcal{S}(r_{n},u_{n})$,

$\displaystyle
I_{2}(s,t)+I_{3}(s,t)\leq\mathrm{cst}\sum_{h=r_{n}+1}^{4r_{n}}g_{n}(h)\leq\mathrm{cst}\sum_{h=r_{n}+1}^{4r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>s_{0}u_{n},\left|\boldsymbol{X}_{h}\right|>s_{0}u_{n})=0\;,$
(5.54)

uniformly in $s,t\in[s_{0},t_{0}]$.

Combination of (5.7.1), (5.53), (5.54) finishes the proof of (5.37).

This, together with (5.36) concluded the proof of (5.34).

###  5.8 Random entropy

In this section we discuss validity of Assumption 3.4. We cannot check this
condition for arbitrary functionals $H$ and anchoring maps $\mathcal{A}$,
however, we will see that the conditions is satisfied for most relevant cases
considered in the paper.

Recall the class

$\mathcal{G}=\\{H^{\mathcal{A}}_{s},s\in[s_{0},t_{0}]\\}=\\{H(\boldsymbol{x}/s)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x}/s)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}},\;\;s\in[s_{0},t_{0}]\\}.$

We start first with $H$ of the form

$\displaystyle
H_{s}(\boldsymbol{x})=\mathbbm{1}{\left\\{K(\boldsymbol{x})>s\right\\}}\;,$
(5.55)

where $K:\mathbb{R}^{\mathbb{Z}}\to\mathbb{R}$. This is the case of the
functionals that lead to the extremal index, the large deviation index and the
ruin index.

Since $\mathcal{A}^{(0)}$ is 0-homogeneous, it does not play a role in
calculating the class entropy. Then

$H_{s}(\boldsymbol{x})\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}}=\mathbbm{1}{\left\\{\min\\{K(\boldsymbol{x}),\left|\boldsymbol{x}_{0}\right|\\}>s\right\\}}\;.$

Hence, the map $s\to
H_{s}(\boldsymbol{x})\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}}$
is decreasing. Therefore, VC($\mathcal{G})=2$.

As for $\mathcal{A}^{(1)}$ we have

$\displaystyle\mathbbm{1}{\left\\{\mathcal{A}^{(1)}(\boldsymbol{x}/s)=0\right\\}}=\mathbbm{1}{\left\\{\boldsymbol{x}_{-\infty,-1}^{*}\leq
s,\left|\boldsymbol{x}_{0}\right|>s\right\\}}\;.$

Thus,

$\displaystyle
H(\boldsymbol{x}/s)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x}/s)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}}=\mathbbm{1}{\left\\{\min\\{K(\boldsymbol{x}),\left|\boldsymbol{x}_{0}\right|\\}>s\right\\}}\mathbbm{1}{\left\\{\boldsymbol{x}_{-\infty,-1}^{*}\leq
s\right\\}}\;.$

Now, the class

$\mathcal{F}=\\{(-\infty,s)\times(s,+\infty):s\in\mathbb{R}\\}$

has the VC-index $3$. By [KS20, Example C.4.14] the class $\mathcal{G}$ has
VC-index at most 3.

Similarly, for $\mathcal{A}^{(2)}$ we have

$\displaystyle\mathbbm{1}{\left\\{\mathcal{A}^{(2)}(\boldsymbol{x}/s)=0\right\\}}=\mathbbm{1}{\left\\{\boldsymbol{x}_{1,\infty}^{*}\leq
s,\left|\boldsymbol{x}_{0}\right|>s\right\\}}\;.$

Thus,

$\displaystyle
H(\boldsymbol{x}/s)\mathbbm{1}{\left\\{\mathcal{A}(\boldsymbol{x}/s)=0\right\\}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{0}\right|>s\right\\}}=\mathbbm{1}{\left\\{\min\\{K(\boldsymbol{x}),\left|\boldsymbol{x}_{0}\right|\\}>s\right\\}}\mathbbm{1}{\left\\{\boldsymbol{x}_{1,\infty}^{*}\leq
s\right\\}}\;$

and again the class $\mathcal{G}$ has VC-index at most 3.

In summary, for functionals $H$ given in (5.55) and the anchoring maps
$\mathcal{A}^{(0)}$, $\mathcal{A}^{(1)}$, $\mathcal{A}^{(2)}$ the class
$\mathcal{G}$ has the VC-index at most 3 and hence the random entropy
Assumption 3.4 is satisfied.

Now, assume that the map $s\to H_{s}$ is decreasing. This is the case of
(again) the extremal index, the large deviation index and the ruin index. This
is also the case of the stop-loss index and the cluster size distribution. If
we choose $\mathcal{A}=\mathcal{A}^{(0)}$, since $\mathcal{A}^{(0)}$ is
$0$-homogeneous, the maps $s\to H_{s}^{\mathcal{A}}$ is also decreasing. Thus,
the VC-index of $\mathcal{G}$ is at most 2. The random entropy condition is
satisfied.

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$\rho=0.9$, Extremal Index=$0.34$ $\rho=0.5$, Extremal Index= $0.94$ ($k$ %)
$k=5$ $k=10$ $k=5$ $k=10$ $r_{n}=8\;$ Disjoint bl 0.360 (0.05) 0.310 (0.03)
0.680 (0.05) 0.570 (0.03) Sliding bl 0.348 (0.04) 0.308 (0.03) 0.673 (0.04)
0.562 (0.03) Runs $C^{(0)}$ 0.240 (0.05) 0.190 (0.03) 0.560 (0.06) 0.420
(0.03) Runs $C^{(1)}$ 0.220 (0.05) 0.170 (0.03) 0.540 (0.06) 0.390 (0.03) Runs
$C^{(2)}$ 0.220 (0.05) 0.170 (0.03) 0.540 (0.06) 0.390 (0.03) $r_{n}=9\;$
Disjoint bl 0.340 (0.05) 0.290 (0.03) 0.655 (0.05) 0.540 (0.03) Sliding bl
0.340 (0.04) 0.309 (0.03) 0.672 (0.04) 0.539 (0.03) Runs $C^{(0)}$ 0.220
(0.05) 0.170 (0.03) 0.520 (0.05) 0.370 (0.03) Runs $C^{(1)}$ 0.200 (0.05)
0.160 (0.03) 0.500 (0.05) 0.340 (0.03) Runs $C^{(2)}$ 0.200 (0.05) 0.160
(0.03) 0.500 (0.05) 0.340 (0.03)

Table 1: The median and the variance (in brackets) of disjoint, sliding blocks
and runs (with the anchoring maps $C^{(0)},C^{(1)}$ and $C^{(2)}$) estimators
for the extremal index. Data are simulated from AR(1) with $\alpha=4$,
$\rho=0.5$ (thus, $\theta=0.94$), and $\rho=0.9$ (thus $\theta=0.34$). Block
size $r_{n}=8,\;9$. The number of order statistics is $k=5\%$ and $10\%$ for a
sample $n=1000$ based on $N=1000$ Monte Carlo simulations.

$\rho=0.9$, Stop-loss Index=$0.085$ $\rho=0.5$, Stop-loss Index= $0.078$ (k %)
$k=10$ $k=40$ $k=10$ $k=40$ $r_{n}=8\;$ Disjoint bl 0.0100 (0.01) 0.0175
(0.007) 0.0500 (0.02) 0.0675 (0.009) Sliding bl 0.0113 (0.01) 0.0181 (0.007)
0.0457 (0.02) 0.0678 (0.008) Runs $C^{(0)}$ 0.0200 (0.01) 0.0250 (0.006)
0.0600 (0.02) 0.0775 (0.007) $r_{n}=9\;$ Disjoint bl 0.0100 (0.01) 0.0200
(0.008) 0.0500 (0.02) 0.0700 (0.009) Sliding bl 0.0122 (0.01) 0.0197 (0.007)
0.0478 (0.02) 0.0694 (0.008) Runs $C^{(0)}$ 0.0200 (0.01) 0.0250 (0.005)
0.0600 (0.02) 0.0750 (0.008)

Table 2: The median and the variance (in brackets) of disjoint, sliding blocks
and runs ($C^{(0)}$) estimators for stop-loss index with $S=0.9$. Data are
simulated from AR(1) with $\alpha=4$, $\rho=0.5,\;=0.9$. The block size is
$r_{n}=8,\;9$. The number of order statistics is $k=10\%$ and $40\%$ for a
sample $n=1000$ based on $N=1000$ Monte Carlo simulations.

Extremal Index=$0.612$ ($k$ %) $k=5$ $k=10$ $r_{n}=8\;$ Disjoint bl 0.660
(0.06) 0.600 (0.04) Sliding bl 0.648 (0.06) 0.593 (0.03) Runs $C^{(0)}$ 0.520
(0.06) 0.410 (0.03) Runs $C^{(1)}$ 0.500 (0.06) 0.380 (0.03) Runs $C^{(2)}$
0.500 (0.06) 0.380 (0.03) $r_{n}=9\;$ Disjoint bl 0.640 (0.06) 0.570 (0.04)
Sliding bl 0.630 (0.06) 0.569 (0.03) Runs $C^{(0)}$ 0.500 (0.06) 0.380 (0.03)
Runs $C^{(1)}$ 0.480 (0.06) 0.340 (0.03) Runs $C^{(2)}$ 0.480 (0.06) 0.340
(0.03)

Table 3: The median and the variance (in brackets) of disjoint, sliding blocks
and runs ($C^{(0)},\;C^{(1)}$ and $C^{(2)}$) estimators for the extremal index
in ARCH(1) model with $\lambda=0.9$. The block size is $r_{n}=8,\;9$. The
number of order statistics is $k=5\%$ and $10\%$ for a sample $n=1000$ based
on $N=1000$ Monte Carlo simulations.