# Limit theorems for unbounded cluster functionals of regularly varying time
series

Zaoli Chen University of Ottawa Rafał Kulik University of Ottawa, email:
<EMAIL_ADDRESS>
###### Abstract

A “blocks method” is used to define clusters of extreme values in stationary
time series. The cluster starts at the first large value in the block and ends
at the last one. The block cluster measure (the point measure at clusters)
encodes different aspects of extremal properties. Its limiting behaviour is
handled by vague convergence, hence the set of test functions consists of
bounded, shift-invariant functionals that vanish around zero. If unbounded or
non shift-invariant functionals are considered, we may obtain convergence at a
different rate, depending on the type of the functional and the block size
(small vs . large blocks). There are two prominent examples of such
functionals: the locations of large jumps and the cluster length. We obtain a
comprehensive characterization of the limiting behaviour of the block cluster
measure evaluated at such functionals for stationary, regularly varying time
series.

Once the convergence of the block cluster measure is established, we can
proceed with consistency of the empirical cluster measure. Consistency holds
in the small and moderate blocks scenario, while fails in the large blocks
situation. Next, we continue with weak convergence of the empirical cluster
processes. The starting point is the seminal paper by Drees and Rootzen
(2010). Under the appropriate uniform integrability condition (related to
small blocks) the results in the latter paper are still valid. In the moderate
and large blocks scenario, the Drees and Rootzen empirical cluster process
diverges, but converges weakly when re-normalized properly.

###### Contents

  1. 1 Introduction
    1. 1.1 Structure of the paper
    2. 1.2 What is missing?
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Tail process
    3. 2.3 Classes of unbounded functions
    4. 2.4 Cluster measure and cluster indices
    5. 2.5 Change of measure
    6. 2.6 Anticlustering conditions
    7. 2.7 Conditional convergence of clusters
    8. 2.8 Vague convergence of cluster measure
    9. 2.9 Dependence assumptions
    10. 2.10 Notation for blocks
      1. 2.10.1 Block size terminology
  3. 3 Convergence of unbounded cluster functionals
    1. 3.1 Summary of the results
    2. 3.2 Location of jumps
      1. 3.2.1 Distribution of the first and the last jump time
      2. 3.2.2 Moments of the first and the last jump time
      3. 3.2.3 Joint moments of the first and the last jump time
    3. 3.3 Functionals dominated by the cluster length
      1. 3.3.1 Distribution of the cluster length
      2. 3.3.2 Moments of the cluster length - small blocks scenario
      3. 3.3.3 Moments of the cluster length - large blocks scenario
  4. 4 Weak convergence of empirical cluster process(es)
    1. 4.1 Functionals dominated by the cluster length
      1. 4.1.1 Small blocks scenario
      2. 4.1.2 Moderate blocks scenario
      3. 4.1.3 Large blocks scenario
    2. 4.2 Location of jumps
  5. 5 Proofs I - unbounded cluster functionals
    1. 5.1 Consequences of the anticlustering conditions
      1. 5.1.1 Formulas for the candidate extremal index
      2. 5.1.2 First and last jump decompositions
      3. 5.1.3 Conditional convergence of clusters
      4. 5.1.4 Vague convergence of clusters
      5. 5.1.5 Proofs for unbounded cluster functionals
      6. 5.1.6 Additional comments for unbounded cluster functionals
  6. 6 Proofs II - the empirical cluster processes

##  1 Introduction

For the last several years, one of the main challenges in extreme value
statistics has been modelling and estimation of clusters of extremes values.
These clusters stem from temporal dependence in time series that represent
real-world phenomena. Cluster indices can be used to quantify different
aspects of clustering of large values. The classical extremal index, the
quantity that appears as an additional parameter in the limiting distribution
of maxima in stationary sequences, is one example of the cluster index. The
extremal index achieves values in $(0,1]$ such that smaller values indicate
stronger clustering.

Cluster indices are estimated by blocks estimators. Early results focused on
asymptotic normality of different types of estimators of the extremal index
including, among others, [16], [7], [8]. Asymptotic results for tail array
sums (see [15]) as well as for cluster size distribution (see [14]) followed.
We note that these papers deal primarily with Peak-over-Threshold (PoT)
framework, as opposed to the block maxima (BM) method (see [3] for a review of
both PoT and BM approaches). The PoT framework is employed in the paper.

The seminal paper [6] provides an unified approach for obtaining limiting
results for clusters of extremes in stationary sequences. The mathematical
set-up of [6] (adapted to the needs of this paper, in the spirit of [9]) is as
follows. We consider a stationary, regularly varying $\mathbb{R}^{d}$-valued
time series $\boldsymbol{X}=\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$. Let
$r_{n}$ be a sequence of nonnegative integers diverging to infinity slower
than $n$. The numbers $r_{n}$ play a role of the block size, needed to capture
clustering of large values. Thus, we have
$m_{n}:=[n/r_{n}]=\max\\{j\in\mathbb{N}:j\leq n/r_{n}\\}$ blocks. Let
$0<u_{n}\to\infty$ and set
$w_{n}:=\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$, where $|\cdot|$ is
a norm on $\mathbb{R}^{d}$. Then

$\displaystyle\mathbb{X}_{j}:=u_{n}^{-1}\left(\boldsymbol{X}_{(j-1)r_{n}+1},\ldots,\boldsymbol{X}_{jr_{n}}\right)\;,\
\ j=1,\ldots,m_{n}\;,$ (1.1)

is the (scaled) $j$th block and

$\displaystyle{\boldsymbol{\nu}}^{*}_{n,r_{n}}:=\frac{1}{r_{n}w_{n}}\mathbb{E}[\delta_{\mathbb{X}_{1}}]$

is the block cluster measure; see Eq. (6.2.1) in [9].

For a given functional $H$ in a suitable function class $\mathcal{H}$, the
cluster index is obtained by evaluating the block cluster measure at $H$. It
is defined as the limit

$\displaystyle{\boldsymbol{\nu}}^{*}(H):=\lim_{n\to\infty}{\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)=\lim_{n\to\infty}\frac{\mathbb{E}[H(\mathbb{X}_{1})]}{r_{n}w_{n}}=\lim_{n\to\infty}\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}\;,\
\ H\in\mathcal{H}\;.$ (1.2)

See [9, Chapter 6]. The rate of convergence $r_{n}w_{n}$ is asymptotically
proportional to the probability of an occurrence of a large value in a single
block. The ”Peak-over-Threshold (POT)” condition (see
$\mathcal{R}(r_{n},u_{n})$ below) yields that $r_{n}w_{n}\to 0$. That is, with
growing $n$, we are less and less likely to observe exceedances over the high
threshold $u_{n}$.

Functionals $H$ are supposed to capture extremal structure of the underlying
time series. Thus, they are defined on $(\mathbb{R}^{d})^{\mathbb{Z}}$ and are
such that their values do not depend on coordinates whose entries are small.
More precisely, for
$\boldsymbol{X}=\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$
and $i\leqslant j\in\mathbb{Z}$, 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. Moreover, we will assume
that $H(\boldsymbol{x})=0$ whenever
$\boldsymbol{x}^{*}:=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|<\epsilon$ for
some $\epsilon>0$. For the latter property, we will say that $H$ vanishes
around zero. Alternatively, we will say that $H$ has support separated from
zero. See Section 2.3 for the precise statement. Some examples of cluster
functionals include:

  * •

$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\boldsymbol{x}^{*}>1\right\\}}$ with
$\boldsymbol{x}\in(\mathbb{R}^{d})^{\mathbb{Z}}$ and
$\boldsymbol{x}^{*}=\sup_{j\in\mathbb{Z}}\left|\boldsymbol{x}_{j}\right|$.
This functional leads to the block estimator of the extremal index.

  * •

$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}=m\right\\}}$
leads to the block estimator of the cluster size distribution (cf. the
relevant estimation problem in [14]).

  * •

$H_{\phi}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\phi(\boldsymbol{x}_{j})$ with
$\phi:\mathbb{R}^{d}\to\mathbb{R}$ gives a tail array sum (cf. [15]). The
choice
$\phi(\boldsymbol{y})=\mathbbm{1}{\left\\{y_{1}>s_{1},\ldots,y_{d}>s_{d}\right\\}}$,
$\boldsymbol{y}=(y_{1},\ldots,y_{d})$, $\boldsymbol{s}=(s_{1},\ldots,s_{d})$,
$s_{i}>0$, leads to the multivariate tail empirical process indexed by
$\boldsymbol{s}$; cf. Example 3.1 in [6].

The appropriate tool to study the limit in (1.2) is vague convergence of
cluster measures. Indeed, Theorem 6.2.5 in [9] (see also Proposition 2.8.1
below) establishes such convergence, where the class $\mathcal{H}$ of test
functions consists of continuous and bounded, shift-invariant functionals that
vanish around zero.

##### Unbounded cluster functionals.

In this paper we want to study general functionals. Besides the natural
mathematical generalisation, there are several motivations for this. A cluster
length, the distance between the location of the last and the first large
value in a block, plays an important role in [6]. Furthermore, it plays a very
special role in the context of asymptotic expansion of blocks estimators, the
problem studied in [4]. Note that the cluster length is shift-invariant, but
unbounded. Hence, at the first step of our analysis we need to extend vague
convergence of the block cluster measure to convergence of unbounded cluster
functionals. Their growth will be controlled by a power of the cluster length
and as such we will refer to them as the cluster length-type functionals. It
turns out that such an extension is non-trivial. Under the appropriate uniform
integrability condition we can recover (1.2) - cluster functionals still have
the rate $r_{n}w_{n}$. The uniform integrability condition holds as long as
the new anticlustering condition is satisfied. The latter in turn is related
to small blocks. On the other hand, in case of large blocks, (1.2) is no
longer true and cluster functionals converge at a different rate.

##### Beyond shift-invariance.

Likewise, for some unbounded non shift-invariant functionals, (1.2) does not
hold for any choice of the blocks size. A particular example here is the
location of large values in a stationary time series.

In summary, in the first step of our analysis we study the limit

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{a_{n}}\;,$

for appropriately chosen functional $H$ and a suitable sequence $a_{n}\to 0$.
The main results that pertain to this problem are as follows.

  * •

Distribution of the locations of jumps in a single block. They are uniformly
distributed and collapse to one random point. The limiting behaviour does not
depend on the block size. See Section 3.2.

  * •

Extension of vague convergence of clusters (Theorem 6.2.5 in [9]) to unbounded
functionals under uniform integrability conditions and phase transition with
respect to block size. See Section 3.3.

##### Consistency of the empirical cluster measure.

An empirical counterpart of ${\boldsymbol{\nu}}^{*}_{n,r_{n}}$ is the
empirical cluster measures

$\displaystyle\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}=\frac{1}{nw_{n}}\sum_{j=1}^{m_{n}}\delta_{\mathbb{X}_{j}}\;;$

cf. Eq. (10.0.1) in [9]. Then, $\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$
can be considered as a disjoint block pseudo-estimator of the cluster index
${\boldsymbol{\nu}}^{*}(H)$. When (1.2) holds, we can expect that
$\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$ is consistent. Indeed, for the
cluster length-type functionals, consistency holds under the small and the
moderate blocks conditions (see (2.12a)-(2.12b)). For the large blocks (see
(2.12c)) the pseudo-estimator diverges. With a different normalization, we can
achieve convergence in probability, but the limit differs from
${\boldsymbol{\nu}}^{*}(H)$. In other words, in the large blocks scenario, we
do not obtain a consistent estimator of the cluster index. See Sections
4.1.1-4.1.3.

Likewise, when $H$ records location of a large jumps, then
$\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$ diverges. On the other hand,
when $H$ is the product of the ”location of a large jump” and a cluster
functional $G$, then after the proper re-normalization we can obtain a
consistent pseudo-estimator of ${\boldsymbol{\nu}}^{*}(G)$. These type of
estimators can help with reducing a downward bias when approximating cluster
indices such as the extremal index. See Section 4.2.

##### Weak convergence of empirical cluster process(es).

The empirical cluster process $\widetilde{\mathbb{G}}_{n}$ is defined as

$\displaystyle\widetilde{\mathbb{G}}_{n}(H)=\frac{1}{\sqrt{nw_{n}}}\sum_{j=1}^{m_{n}}\left\\{H(\mathbb{X}_{j})-\mathbb{E}[H(\mathbb{X}_{j})]\right\\}\;,$
(1.3)

where $H$ belongs to a suitably chosen function class. The functional central
limit theorem for the empirical cluster process is established in [6]. In the
context of regularly varying time series the result is given in [9]; see
Theorem 10.2.1 there.

For the cluster length-type functionals, the empirical cluster process
$\widetilde{\mathbb{G}}_{n}$ converges in the small blocks scenario, while it
diverges when moderate blocks are considered. We need to modify the empirical
cluster process. Recall that in both scenarios, the pseudo-estimator is
consistent. See Sections 4.1.1-4.1.2.

In case of functionals that record jump locations, again a modified empirical
cluster process has to be introduced. See Section 4.2.

##### Statistical consequences.

The take out message of this paper is that the ”classical” theory of weak
convergence of clusters (as introduced in [6]) may fail in case of unbounded
functionals. The asymptotics may depend on the type of the functional and the
block size. Large blocks should be avoided, since they may lead to either
different central limit theorem or inconsistency of blocks estimators.

On the other hand, the blocks estimators based on unbounded functionals may
serve as alternative way of estimating cluster indices. In particular, they
may help with a downward bias when estimating the classical extremal index.
See Example 4.2.2.

##### Block size terminology.

For convergence of cluster functionals, we distinguish between small and large
blocks. The notion small/large is relative; it depends on the type of
functional. For consistency and weak convergence, we have to consider small,
moderate and large blocks. Again, these notions are relative. See Section
2.10.1.

###  1.1 Structure of the paper

Section 2 consists preliminaries. It introduces the tail process, the relevant
class of functions (including the cluster length, which appears to be the most
important functional in the context of the paper; see Section 2.3), cluster
measures and cluster indices. Section 2.6 introduces different types of
anticlustering conditions. Section 2.8 recalls vague convergence of cluster
measures. Proposition 2.8.1 is the most relevant result in this context.

In Section 3 we state a variety of results for unbounded cluster functionals.
Locations of jumps (Section 3.2) and functionals dominated by the cluster
length (Section 3.3) play a special role. In the second case, we have to
distinguish between small and large blocks. The entire contents of this
section is new.

Section 4 deals with consistency of blocks estimators and weak convergence of
the corresponding empirical cluster processes. The results there are new,
however, their proofs follow in a relatively straightforward way from [9,
Chapter 10].

Sections 5 and 6 contain all the proofs for unbounded functionals and weak
convergence of the empirical cluster processes, respectively.

The goal of Sections 3 and 4 is to state clearly the results, without too much
additional ”noise”. A variety of additional remarks can be found in Section 5.

###  1.2 What is missing?

We provide a comprehensive theory in the small blocks scenario. The behaviour
in the large blocks situation is limited to some specific functionals and
under rather restrictive dependence assumptions. This can be viewed as a
limitation of the paper, however, one can argue that large blocks scenario is
much less relevant from the statistical perspective.

Also, we do not consider here functionals that do not vanish around zero, for
example those that lead to the large deviation index (see e.g. [11]). These
functionals may be large due to a cumulation of small values. This may be
prevented by imposing a ”negligibility of small values” condition, but it is
still not suitable for the techniques used in the paper.

##  2 Preliminaries

In this section we fix the notation and introduce the relevant classes of
functions. In Section 2.2 we recall the notion of the tail and the spectral
tail process (cf. [2]). In Section 2.4 we define cluster indices; see [9,
Chapter 5] for a detailed introduction.

In Section 2.6 we introduce anticlustering conditions. The first one, the
classical $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is needed, to establish
convergence of the cluster measure; see Section 2.8. The results of the latter
section are extracted from [9, Chapter 6]. See also [13] and [1]. Another
condition, called here $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$, is crucial for
many results in the paper.

In Section 2.9 we introduce our dependence assumptions.

###  2.1 Notation

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

$\displaystyle\lim_{n\to\infty}u_{n}=\lim_{n\to\infty}r_{n}=\lim_{n\to\infty}nw_{n}=\infty\;,\
\ \lim_{n\to\infty}r_{n}/n=\lim_{n\to\infty}r_{n}w_{n}=0\;,$
($\mathcal{R}(r_{n},u_{n})$)

where $w_{n}=\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n})$. For a
sequence
$\boldsymbol{x}=\\{\boldsymbol{x}_{j},j\in\mathbb{Z}\\}\in(\mathbb{R}^{d})^{\mathbb{Z}}$
and $i\leqslant j\in\mathbb{Z}\cup\\{-\infty,\infty\\}$ denote
$\boldsymbol{x}_{i,j}=(\boldsymbol{x}_{i},\ldots,\boldsymbol{x}_{j})\in(\mathbb{R}^{d})^{j-i+1}$,
$\boldsymbol{x}_{i,j}^{\ast}=\max_{i\leqslant l\leqslant
j}|\boldsymbol{x}_{l}|$ and
$\boldsymbol{x}^{\ast}=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|$. By
${\boldsymbol{0}}$ we denote the zero sequence; its dimension can be different
at 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.

###  2.2 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
$\alpha$. 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 $i\leqslant j\in\mathbb{Z}$. We call $\boldsymbol{Y}$
the tail process. See [2]. 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 law of the tail process $\boldsymbol{Y}$,

$\displaystyle\lim_{x\to\infty}\frac{\mathbb{E}[H(x^{-1}\boldsymbol{X})\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{0}\right|>x\right\\}}]}{\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>x)}$
$\displaystyle=\mathbb{E}[H(\boldsymbol{Y})]\;.$ (2.1)

The random variable $|\boldsymbol{Y}_{0}|$ has the Pareto distribution with
index $\alpha$ and hence $\left|\boldsymbol{Y}_{0}\right|>1$. The spectral
tail process
$\boldsymbol{\Theta}=\\{\boldsymbol{\Theta}_{j},j\in\mathbb{Z}\\}$ is defined
as $\boldsymbol{\Theta}_{j}=\boldsymbol{Y}_{j}/|\boldsymbol{Y}_{0}|$.

###  2.3 Classes of unbounded functions

Let $\epsilon>0$. Define

$\displaystyle T^{(1)}(\boldsymbol{x},\epsilon)=T_{\rm
min}(\boldsymbol{x},\epsilon)$
$\displaystyle=\inf\\{j\in\mathbb{Z}:|\boldsymbol{x}_{j}|>\epsilon\\}\;,$
(2.2a) $\displaystyle T_{\rm max}(\boldsymbol{x},\epsilon)$
$\displaystyle=\sup\\{j\in\mathbb{Z}:|\boldsymbol{x}_{j}|>\epsilon\\}\;,$
(2.2b) $\displaystyle T^{(i+1)}(\boldsymbol{x},\epsilon)$
$\displaystyle=\inf\\{j>T^{(i)}(\boldsymbol{x},\epsilon):|\boldsymbol{x}_{j}|>\epsilon\\}\;,\
\ i\geqslant 1\;.$ (2.2c)

We will use the convention $\inf\\{\emptyset\\}=+\infty$,
$\sup\\{\emptyset\\}=-\infty$. If $\boldsymbol{x}\in\ell_{0}(\mathbb{R}^{d})$,
then $\sup\\{j\geqslant 0:|\boldsymbol{x}_{j}|>\epsilon\\}<\infty$ and
$\inf\\{j\geqslant 0:|\boldsymbol{x}_{j}|>\epsilon\\}>-\infty$. Hence, when
restricted to $\ell_{0}$, the functionals $T^{(i)}$, $i\geqslant 1$, and
$T_{\rm max}$ attain finite values.

Formally, the above functionals depend on $\epsilon$. However, without loss of
generality we will assume throughout the paper that $\epsilon=1$ and we will
use the notation $T^{(i)}(\boldsymbol{x})$ instead.

Let again $\boldsymbol{x}\in\ell_{0}(\mathbb{R}^{d})$. We can define the
cluster length functional as

$\displaystyle\mathcal{L}(\boldsymbol{x})=T_{\rm max}(\boldsymbol{x})-T_{\rm
min}(\boldsymbol{x})+1\;$ (2.3)

with the convention $\mathcal{L}(\boldsymbol{x})=0$ whenever
$\boldsymbol{x}^{\ast}=\sup_{j\in\mathbb{Z}}\left|\boldsymbol{x}_{j}\right|\leqslant
1$. We note also that if $\left|\boldsymbol{x}_{0}\right|>1$, while
$\sup_{j\in\mathbb{Z},j\not=0}\left|\boldsymbol{x}_{j}\right|\leqslant 1$,
then $\mathcal{L}(\boldsymbol{x})=1$.

We define further the functional $\mathcal{E}$ by
$\mathcal{E}(\boldsymbol{x})=\sum_{j\in\mathbb{Z}}\mathbbm{1}{\left\\{\left|\boldsymbol{x}_{j}\right|>1\right\\}}$.
It returns the number of exceedances over 1.

We shall consider functionals under the following assumptions.

######  Assumption 2.3.1.

We denote by $\mathcal{H}$ a collection of functionals
$\ell_{0}(\mathbb{R}^{d})\to\mathbb{R}_{+}$ such that each $H\in\mathcal{H}$
satisfies:

  * $(\textup{i})$

$H$ is continuous with respect to the law of the process $\boldsymbol{Y}$;

  * $(\textup{ii})$

If $\mathcal{E}(\boldsymbol{x})=0$, then $H(\boldsymbol{x})=0$;

  * $(\textup{iii})$

If $\mathcal{E}(\boldsymbol{x})>0$, then
$H(\boldsymbol{x})=H(\boldsymbol{x}_{T_{\rm min}(\boldsymbol{x}),T_{\rm
max}(\boldsymbol{x})})$, where $T_{\rm min}(\boldsymbol{x})$ and $T_{\rm
max}(\boldsymbol{x})$, are the first and the last exceedance times defined in
(2.2a)-(2.2b).

We note that the above assumption allows for unbounded functionals. We will
need to control their growth.

######  Assumption 2.3.2.

We denote by $\mathcal{H}(\gamma)\subseteq\mathcal{H}$ a collection of
functionals $\ell_{0}(\mathbb{R}^{d})\to\mathbb{R}$ such that each
$H\in\mathcal{H}(\gamma)$ satisfies:

  * $(\textup{iv})$

There exists a constant $C_{H}>0$ such that $H(\boldsymbol{x})\leqslant
C_{H}\big{[}\mathcal{L}(\boldsymbol{x})\big{]}^{\gamma}$ for all
$\boldsymbol{x}\in\ell_{0}(\mathbb{R}^{d})$.

######  Example 2.3.3.

Obviously, we can take $H$ as the cluster length functional itself:
$H=\mathcal{L}$. Then $\gamma=1$.

The extremal index functional defined as
$\mathcal{T}(\boldsymbol{x})=\mathbbm{1}{\left\\{\boldsymbol{x}^{\ast}>1\right\\}}$.
It fulfills Assumption 2.3.2 with $C_{\mathcal{T}}=1$ and $\gamma=0$. This
functional is bounded. In particular, $\mathcal{T}(\boldsymbol{Y})=1$.

In case of extremal independence, $\left|\boldsymbol{Y}_{j}\right|=0$ for
$|j|\geqslant 1$ and hence
$\mathcal{T}(\boldsymbol{Y})=\mathcal{L}(\boldsymbol{Y})=1$.

######  Remark 2.1.

Functional can depend on small values, but only those that occur between large
values (”within a cluster”). For example, $H_{\rm
sum}(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j=T_{\rm
min}(\boldsymbol{x})}^{T_{\rm
max}(\boldsymbol{x})}|\boldsymbol{x}_{j}|>\eta\right\\}}$ (with $\eta>0$)
fulfills Assumption 2.3.1, but
$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\sum_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|>1\right\\}}$
does not. The latter functional appears in the context of large deviations,
see [11].

We also note that the functionals $T^{(i)}$ defined in (2.2c) do not fulfill
Assumption 2.3.1(iii). Furthermore, they cannot be bounded by the cluster
length, and hence they do not fulfill Assumption 2.3.2.

The class $\mathcal{H}(\gamma)$ is parametrized by $\gamma$ that will play an
important role. The case $\gamma=0$ corresponds to bounded functionals. If a
function $H$ is bounded, we will denote
$\|H\|=\sup_{\boldsymbol{x}\in(\mathbb{R}^{d})^{\mathbb{Z}}}|H(\boldsymbol{x})|$.
The property $(\textup{ii})$ will be referred to as ”$H$ vanishes around
${\boldsymbol{0}}$” or ”$H$ has support separated from ${\boldsymbol{0}}$”. In
particular, $\mathcal{L}$, $\mathcal{T}$ and the aforementioned $H_{\rm sum}$
vanish around ${\boldsymbol{0}}$. However, the large deviation functional does
not vanish around ${\boldsymbol{0}}$.

###  2.4 Cluster measure and cluster indices

Consider the infargmax functional $\mathcal{A}_{0}$ defined on
$(\mathbb{R}^{d})^{\mathbb{Z}}$ by
$\mathcal{A}_{0}(\boldsymbol{x})=\inf\\{j:\boldsymbol{x}_{-\infty,{j}}^{\ast}=\boldsymbol{x}^{\ast}\\}$,
with the convention that $\inf\\{\emptyset\\}=+\infty$. If
$\mathbb{P}(\mathcal{A}_{0}(\boldsymbol{Y})\notin\mathbb{Z})=0$ then we can
define

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

In fact, $\mathcal{A}_{0}$ can be replaced with any anchoring map (see [9,
Theorem 5.4.2]). Hence,

$\displaystyle{\vartheta}=\mathbb{P}(\mathcal{A}_{0}(\boldsymbol{Y})=0)=\mathbb{P}\left(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1\right)=\mathbb{P}\left(\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant
1\right)\;.$ (2.4)

Therefore, ${\vartheta}$ can be recognized as the (candidate) extremal index.
It becomes the usual extremal index under additional mixing and anticlustering
conditions.

######  Definition 2.4.1 (Cluster measure).

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

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

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. For every bounded or non-
negative functional $H$ such that $H(\boldsymbol{x})=0$ if
$\boldsymbol{x}^{\ast}\leqslant 1$ we have

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

######  Definition 2.4.2 (Cluster index).

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

###  2.5 Change of measure

It is important to notice the presence of
$\mathbbm{1}{\left\\{\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant 1\right\\}}$
in the definition of ${\boldsymbol{\nu}}^{*}$. It will be convenient to
express some formulas in the language of an auxiliary process
$\boldsymbol{Z}$. Assume that
$\mathbb{P}(\lim_{|j|\to\infty}|\boldsymbol{Y}_{j}|=0)=1$ and define the
process $\boldsymbol{Z}$ as $\boldsymbol{Y}$ conditioned on the first
exceedance over 1 to happen at time zero, that is
$\boldsymbol{Y}_{-\infty,-1}^{*}\leqslant 1$. Then, combining (2.4) with
(2.5), we obtain

$\displaystyle{\boldsymbol{\nu}}^{*}(H)=\mathbb{E}[H(\boldsymbol{Y})\mathbbm{1}{\left\\{\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1\right\\}}]={\vartheta}\mathbb{E}[H(\boldsymbol{Z})]\;.$

Informally speaking, the distribution of $\boldsymbol{Z}$ is a Palm version of
the distribution of $\boldsymbol{Y}$. See [12] and [10] for a relation between
the tail process and Palm theory. See also Exercise 5.29 in [9].

###  2.6 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 $\boldsymbol{Y}_{-r,r}$. 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 that controls persistence of large values on one block.

######  Definition 2.6.1 ([5], Condition 2.8).

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

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

Condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is referred to as the (basic)
anticlustering condition. $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ implies that
$\boldsymbol{Y}\in\ell_{0}(\mathbb{R}^{d})$ and
$\vartheta=\mathbb{P}\left(\boldsymbol{Y}^{\ast}_{-\infty,-1}\leqslant
1\right)>0$. Also, $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds for sequence of
i.i.d. random variables whenever $\lim_{n\to\infty}r_{n}w_{n}=0$.

######  Definition 2.6.2.

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

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

This condition is new. It is an extension of the existing condition
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ with $\gamma=0$. The latter was proven
to hold for a large class of time series models, such as geometrically ergodic
Markov chains, short-memory linear or max-stable processes. See [9].

######  Remark 2.2.

  * •

The condition implies that
$\sum_{i\in\mathbb{Z}}i^{\gamma}\mathbb{P}(\left|\boldsymbol{Y}_{i}\right|>1)<\infty$.
See Lemma 9.2.3 in [9] in case $\gamma=0$. Its proof can be easily modified to
arbitrary $\gamma>0$.

  * •

It is obvious that in case of i.i.d. or $\ell$-dependent sequences
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ holds if and only if
$\lim_{n\to\infty}r_{n}^{\gamma+1}w_{n}=0$. However, we could not establish
that $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ and
$\lim_{n\to\infty}r_{n}^{\gamma+1}w_{n}=0$ are equivalent in general.

  * •

The condition $\mathcal{S}^{(\gamma_{1})}(r_{n},u_{n})$ implies
$\mathcal{S}^{(\gamma_{2})}(r_{n},u_{n})$ whenever
$\gamma_{1}>\gamma_{2}\geqslant 0$. The condition
$\mathcal{S}^{(0)}(r_{n},u_{n})$ implies the condition
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$.

###  2.7 Conditional convergence of clusters

From the definition of the tail process we obtain immediately

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

for $i\leqslant j$ and suitable functionals $H$. Thanks to the anticlustering
condition we can replace $i,j$ with a sequence diverging to $\infty$.

######  Proposition 2.7.1 ([2], Proposition 4.2; [9], Theorem 6.1.4).

Let $\boldsymbol{X}$ be a stationary time series. Assume that
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let $H$ be a bounded or non-
negative functional defined on $\ell_{0}(\mathbb{R}^{d})$ that is almost
surely continuous with respect to the distribution of the tail process
$\boldsymbol{Y}$. Then for any constant $C>0$,

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

###  2.8 Vague convergence of cluster measure

We now investigate the unconditional convergence of
$u_{n}^{-1}\boldsymbol{X}_{1,r_{n}}$. Contrary to Proposition 2.7.1, where an
extreme value was imposed at time 0, a large value in the cluster can happen
at any time.

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

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

We are interested in convergence of ${\boldsymbol{\nu}}^{*}_{n,r_{n}}$ to
${\boldsymbol{\nu}}^{*}$. We quote Theorem 6.2.5 in [9].

######  Proposition 2.8.1.

Let condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ hold. Then, for all
bounded continuous shift-invariant functionals $H$ with support separated from
${\boldsymbol{0}}$ we have

$\displaystyle\lim_{n\to\infty}{\boldsymbol{\nu}}^{*}_{n,r_{n}}(H)=\lim_{n\to\infty}\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}=\boldsymbol{\nu}^{\ast}(H)\;.$
(2.6)

The immediate consequence is the following limit:

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{P}(\boldsymbol{X}_{1,r_{n}}^{\ast}>u_{n})}{r_{n}w_{n}}={\vartheta}\;.$
(2.7)

If $H$ does not fulfill the conditions of Proposition 2.8.1, we may have
several scenarios, as studied in Section 3.

###  2.9 Dependence assumptions

We shall divide the sample
$\\{\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\\}$ into $m_{n}\in\mathbb{N}$
disjoint blocks of same size $r_{n}\in\mathbb{N}$. Without loss of generality
we will assume that $n=m_{n}r_{n}$. Dependence within each block (”local
dependence”) is controlled by the appropriate anticlustering condition.
Dependence between the blocks (”global dependence”) is controlled by
$\beta$-mixing. We will assume that there exists a sequence $\ell_{n}$ such
that the following condition holds:

$\displaystyle\lim_{n\to\infty}\frac{1}{\ell_{n}}=\lim_{n\to\infty}\frac{\ell_{n}}{r_{n}}=\lim_{n\to\infty}\frac{r_{n}}{n}=\lim_{n\to\infty}\frac{n}{r_{n}}\beta_{\ell_{n}}=0\;.$
($\beta(r_{n},\ell_{n})$)

We recall the classical covariance inequality for bounded, beta-mixing random
variables. Let $\beta({\mathcal{F}_{1}},{\mathcal{F}_{2}})$ be the
$\beta$-mixing coefficient between two sigma fields. Then

$\displaystyle|\mathrm{Cov}(H(Z_{1}),H(Z_{2}))|\leq\mathrm{cst}\
\|H\|_{\infty}\|\widetilde{H}\|_{\infty}\beta(\sigma(Z_{1}),\sigma(Z_{2}))\;.$
(2.8)

In (2.8) the constant $\mathrm{cst}$ does not depend on $H,\widetilde{H}$.

###  2.10 Notation for blocks

We consider disjoint blocks of size $r_{n}$:

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

The subscript $j$ will always indicate the numbering of blocks. Denote by

$\displaystyle
N_{j}:=\mathcal{E}(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})=\sum_{i\in
I_{j}}\mathbbm{1}{\left\\{\left|\boldsymbol{X}_{i}\right|>u_{n}\right\\}}\;,\
\ j=1,\ldots,m_{n}\;,$

the number of exceedances in the block $I_{j}$. Set

$\displaystyle A_{j}=\\{\exists i:(j-1)r_{n}+1\leqslant i\leqslant
jr_{n},|\boldsymbol{X}_{i}|>u_{n}\\}=\\{\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}}^{\ast}>u_{n}\\}\;.$
(2.9)

Recall the notation (2.2a)-(2.2c). If $N_{j}\not=0$, then denote by

$\displaystyle
t_{j}{(i)}=T^{(i)}(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})$

the $i$th exceedance time in the block $j$. Hence,

$\displaystyle t_{j}{(1)}<\cdots<t_{j}{(N_{j})}\;,\ \ j=1,\ldots,m_{n}\;,$
(2.10)

are the exceedance times in the $j$th block. We will use the convention
$t_{j}(0)\equiv(j-1)r_{n}$ and $t_{j}(N_{j}+1)\equiv jr_{n}$. Note that it is
possible that $t_{j}{(N_{j})}=t_{j}(N_{j}+1)=jr_{n}$. This happens when the
last jump in the block $j$ occurs at the right-end point, $jr_{n}$. On the
other hand, since the $j$th block starts at $(j-1)r_{n}+1$, $t_{j}(1)$ is
strictly larger than $t_{j}(0)$. We will set $t_{1}(i)=t(i)$ for
$i=1,\ldots,N_{1}$.

Furthermore, using the notation introduced in (2.3),

$\displaystyle
t_{j}{(N_{j})}-t_{j}{(1)}+1=\mathcal{L}(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})=:\mathcal{L}_{j}$
(2.11)

is the cluster length in the $j$th block. We use the convention
$\sum_{i=\ell}^{j}=0$ whenever $j<\ell$. Hence, $\mathcal{L}_{j}=1$ whenever
$N_{j}=1$. If there are no exceedances over the threshold $u_{n}$, then we set
$\mathcal{L}_{j}\equiv 0$ (there will be no issue with this, since
$\mathcal{L}_{j}$ will appear with the appropriate indicator.) Also, by the
definition, $\mathcal{L}_{j}\leqslant r_{n}$ uniformly in $j$.

We will also use the following notation for the scaled block $j$:

$\displaystyle\mathbb{X}_{j}=u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}}\;,\
\ j=1,\ldots,m_{n}\;.$

####  2.10.1 Block size terminology

In what follows, when convergence of cluster functionals is considered, the
small and large blocks refers to $r_{n}^{\gamma+1}w_{n}\to 0$ and
$r_{n}^{\gamma+1}w_{n}\to\infty$, respectively. In the context of weak
convergence of empirical processes, we will distinguish between three
different scenarios, referred to as the small (respectively, moderate or
large) blocks condition for empirical processes:

$\displaystyle r_{n}^{2\gamma+1}w_{n}\to 0\ \ $ $\displaystyle{\rm(small\
blocks)}\;,$ (2.12a) $\displaystyle r_{n}^{\gamma+1}w_{n}\to 0\ {\rm and}\
r_{n}^{2\gamma+1}w_{n}\to\infty\ \ $ $\displaystyle{\rm(moderate\ blocks)}\;,$
(2.12b) $\displaystyle r_{n}^{\gamma+1}w_{n}\to\infty\ \ $
$\displaystyle{\rm(large\ blocks)}\;.$ (2.12c)

##  3 Convergence of unbounded cluster functionals

In what follows, $\boldsymbol{X}$ is $\mathbb{R}^{d}$-valued regularly varying
time series. Also, $\mathcal{R}(r_{n},u_{n})$ is assumed everywhere.

Proposition 2.8.1 is valid for bounded, shift-invariant functionals that
vanish around zero. In other words, one deals with convergence of the
appropriately scaled
$\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{A_{1}\right\\}}]$,
where $A_{1}=\\{\boldsymbol{X}_{1,r_{n}}^{\ast}>u_{n}\\}$. The goal is to
extend it to unbounded functionals.

Set $t(i)=t_{1}(i)=T^{(i)}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})$,
$i=1,\ldots,N_{1}$. Hence, $t(1)=T_{\min}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})$
and $t{(N_{1})}=T_{\max}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})$.

###  3.1 Summary of the results

  * •

Section 3.2: The functionals $T_{\min}$ and $T_{\max}$ are unbounded and not
shift-invariant. Then, $t(1)$ and $t(N_{1})$ are not tight under the
conditional law (given $A_{1}$) - they need to be scaled by the block size
$r_{n}$ and the limiting conditional distribution is uniform. See Lemma 3.2.1.
Next, Proposition 2.8.1 is no longer valid - the $\gamma$ moment of $t(1)$ or
$t(N_{1})$ requires the scaling $r_{n}^{\gamma+1}w_{n}$ instead of
$r_{n}w_{n}$. See Lemma 3.2.2.

  * •

Section 3.3 deals with functionals bounded by the cluster length. Even though
the cluster length is a simple function of $t({1})$ and $t(N_{1})$, its
behaviour changes significantly. First, the cluster length is tight under the
conditional law; see Lemma 3.3.2. Next, in the small blocks scenario the
uniform integrability holds and Proposition 2.8.1 is still valid. It fails in
the large blocks scenario. See Corollary 3.3.4 and Lemma 3.3.5. For the
$\gamma$ moment of the cluster length, the dichotomous between the small and
the large blocks scenario is related to whether $r_{n}^{\gamma+1}w_{n}\to 0$
or $r_{n}^{\gamma+1}w_{n}\to\infty$.

All proofs are given in Section 5.1.5.

###  3.2 Location of jumps

First, we analyse the limiting conditional distribution of the first and the
last jump times. Note that these functionals are neither bounded nor shift-
invariant. When scaled by the block size, both become uniformly distributed.
The uniform limiting conditional distribution yields immediately the
convergence of conditional moments. The main results are: Lemma 3.2.1 Lemma
3.2.2 and Corollary 3.2.3. No specific dependence structure is needed; only
the basic anticlustering condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is
used. Next, we deal with joint moments of the first and the last jump. The
main result is Lemma 3.2.4. There, we need a slightly stronger anticlustering
condition $\mathcal{S}^{(0)}(r_{n},u_{n})$. The results there are quite
intuitive: the first and the last jump time have to be scaled by the block
size $r_{n}$, while the presence of the indicator of a large jump forces the
additional $r_{n}w_{n}$. See Corollary 3.2.3. It has to be emphasized that the
scaling is the same for any choice of the block size.

In summary, the results of this section show that Proposition 2.8.1 is no
longer valid for the unbounded, non shift-invariant functionals $T_{\min}$ and
$T_{\max}$.

####  3.2.1 Distribution of the first and the last jump time

The first result shows that the scaled jump times $t(1)$ and $t(N_{1})$ are
approximately uniformly distributed. Only the basic anticlustering condition
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$ is needed.

######  Lemma 3.2.1.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let $q\in(0,1)$.

  1. (i)

We have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}\mathbb{P}(\\{t(1)>qr_{n}\\}\cap
A_{1})=\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}\mathbb{P}(\\{t(N_{1})>qr_{n}\\}\cap
A_{1})=(1-q){\vartheta}\;.$

  2. (ii)

We have

$\displaystyle\frac{\mathbb{E}\left[\mathbbm{1}{\left\\{qr_{n}<t(1)<r_{n}\right\\}}G\left(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}}\right)\right]}{r_{n}w_{n}}\to(1-q){\boldsymbol{\nu}}^{*}(G)\;.$

for any bounded measurable functional
$G:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$ that is continuous with respect
to the law of $\boldsymbol{Y}$, with support separated from
${\boldsymbol{0}}$.

Since $\mathbb{P}(A_{1})\sim{\vartheta}r_{n}w_{n}$ (see (2.7)), we can
rephrase the above result as:

$\displaystyle\lim_{n\to\infty}\mathbb{P}(\\{t(1)>qr_{n}\\}\mid A_{1})=1-q\;.$

####  3.2.2 Moments of the first and the last jump time

The next statements are almost obvious in view of the limiting conditional
distribution of $t(1)/r_{n}$ being standard uniform.

######  Lemma 3.2.2.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be $\gamma$-homogenous, $\gamma>0$. Then

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[f(t(1))G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{\gamma+1}w_{n}}=\lim_{n\to\infty}\frac{\mathbb{E}[f(t(N_{1}))G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{\gamma+1}w_{n}}=\frac{f(1)}{\gamma+1}{\boldsymbol{\nu}}^{*}(G)\;$

for any bounded measurable functional
$G:(\mathbb{R}^{d})^{\mathbb{Z}}\to\mathbb{R}$ that is continuous with respect
to the law of $\boldsymbol{Y}$, with support separated from
${\boldsymbol{0}}$.

######  Corollary 3.2.3.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let $\gamma>0$. Then

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[t(1)^{\gamma}\mathbbm{1}{\left\\{A_{1}\right\\}}]}{r_{n}^{\gamma+1}w_{n}}=\lim_{n\to\infty}\frac{\mathbb{E}[(r_{n}-t(1))^{\gamma}\mathbbm{1}{\left\\{A_{1}\right\\}}]}{r_{n}^{\gamma+1}w_{n}}=\frac{1}{\gamma+1}{\vartheta}\;,$

with the corresponding statements for $t(N_{1})$.

####  3.2.3 Joint moments of the first and the last jump time

To study joint moments we need a slightly stronger anticlustering condition
$\mathcal{S}^{(0)}(r_{n},u_{n})$. Also, a function
$f:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$ is
$(\gamma_{1},\gamma_{2})$-homogenous (with $\gamma_{1},\gamma_{2}\geqslant 0$)
if for all $a,b,s,t>0$, $f(as,bt)=a^{\gamma_{1}}b^{\gamma_{2}}f(s,t)$.

######  Lemma 3.2.4.

Assume that $\mathcal{S}^{(0)}(r_{n},u_{n})$ holds. Let
$f:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$ be
$(\gamma_{1},\gamma_{2})$-homogenous, $\gamma_{1},\gamma_{2}\geqslant 0$. Then

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma_{1}+\gamma_{2}+1}w_{n}}\mathbb{E}[f\left(t(1),t(N_{1})\right)\mathbbm{1}{\left\\{A_{1}\right\\}}]=\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}{\vartheta}\;.$
(3.1)

###  3.3 Functionals dominated by the cluster length

Note first that the cluster length functional defined in (2.3) is unbounded,
but shift-invariant. Recall the definition (2.11) of the cluster length:
$\mathcal{L}_{j}=t_{j}{(N_{j})}-t_{j}{(1)}+1$.

We continue with an extension of Proposition 2.8.1. We start with the
following obvious result. We replace boundedness with the uniform
integrability condition.

######  Proposition 3.3.1.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Let $H$ be a
continuous shift-invariant functional with support separated from
${\boldsymbol{0}}$. Assume that

$\displaystyle\lim_{\ell\to\infty}\limsup_{n\to\infty}\frac{\mathbb{E}[|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})>\ell\right\\}}]}{r_{n}w_{n}}=0\;.$
(3.2)

Then (2.6) holds.

In this context, the main functional of interest will be a cluster length or
any functional bounded by (a power of) the cluster length.

Recall Lemma 3.2.4. Since $\mathcal{L}_{1}-1=t(N_{1})-t(1)$, one could expect
that the limiting behaviour of the cluster length follows directly from that
of the jump locations. However, we notice that the right of (3.1) vanishes for
the function $f(x,y)=(y-x)_{+}$. Indeed, the cluster length behaviour differs
significantly. Lemma 3.3.2 (that follows directly from the vague convergence
in Proposition 2.8.1) gives tightness of the limiting conditional distribution
of the (unscaled) cluster length. This is in contrast to the first and the
last jump locations, that have to be scaled by the block size.

We then extend Proposition 2.8.1 to the moments of the cluster length.
Contrary to the jump times, there is a phase transition: for the first moment,
the scaling is either $r_{n}w_{n}$ or $r_{n}^{3}w_{n}^{2}$ depending on
whether the blocks are small ($r_{n}^{2}w_{n}\to 0$) or large
($r_{n}^{2}w_{n}\to\infty$). In the case of the small blocks it is a
consequence of vague convergence in Proposition 2.8.1 and the uniform
integrability. See Corollary 3.3.4. Here, a stronger anticlustering condition
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ is needed. We discuss its optimality in
Remark 5.3. Lemma 3.3.5 covers the case of large blocks. As expected, a
different scaling is needed.

####  3.3.1 Distribution of the cluster length

The following result is a re-statement of the one in [6] under weaker
assumptions (see Lemma 5.2 and Eq. (5.19) there). Only the anticlustering
condition is needed here. The result follows directly from Proposition 2.8.1.

######  Lemma 3.3.2.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds. Then for any finite
$q\in\\{0,\ldots,r_{n}\\}$,

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}\mathbb{P}\left(\\{\mathcal{L}_{1}>q\\}\cap
A_{1}\right)={\boldsymbol{\nu}}^{*}(\mathbbm{1}{\left\\{\mathcal{L}>q\right\\}})\;.$
(3.3)

In particular, the cluster length $\mathcal{L}_{1}$ is tight under the
conditional law $\mathbb{P}(\cdot\mid A_{1})$.

We note that we can re-write (3.3) in several equivalent ways:

$\displaystyle{\boldsymbol{\nu}}^{*}(\mathbbm{1}{\left\\{\mathcal{L}>q\right\\}})=\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\mathcal{L}(\boldsymbol{Y})>q)=\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{q,\infty}^{\ast}>1)={\vartheta}\mathbb{P}(\mathcal{L}(\boldsymbol{Z})>q)\;.$

As will argue in Remark 5.4, the limiting distribution of the cluster length
is a proper distribution.

####  3.3.2 Moments of the cluster length - small blocks scenario

Lemma 3.2.4 describes the limiting behaviour of the jump times. However, the
asymptotics for the moments of $(t(N_{1})-t(1))$ changes completely. In order
to proceed, we need the stronger version of the anticlustering condition.
Recall that $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ implies
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$. Also, recall that
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ forces the blocks to be small, since in
the i.i.d. case the anticlustering is equivalent to $r_{n}^{\gamma+1}w_{n}\to
0$. As seen in Remark 5.3 below, the latter small blocks condition is
necessary for convergence of the moments of the cluster length.

The main result in case of small blocks is Corollary 3.3.4. It follows from
vague convergence along with the uniform integrability. Lemma 3.3.5 covers the
case of large blocks.

######  Lemma 3.3.3.

Assume that $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ holds. Let
$H\in\mathcal{H}(\gamma)$. Then the uniform integrability condition (3.2)
holds.

We obtain immediately the following corollary.

######  Corollary 3.3.4.

Assume that $\mathcal{S}^{(\gamma+\delta)}(r_{n},u_{n})$ holds. Then for any
$G\in\mathcal{H}(\delta)$,

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}\mathbb{E}\left[\mathcal{L}^{\gamma}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{A_{1}\right\\}}\right]={\boldsymbol{\nu}}^{*}(G\mathcal{L}^{\gamma})={\vartheta}\mathbb{E}\left[G(\boldsymbol{Z})\mathcal{L}^{\gamma}(\boldsymbol{Z})\right]\;.$

####  3.3.3 Moments of the cluster length - large blocks scenario

We recall that $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ is almost equivalent to
$r_{n}^{\gamma+1}w_{n}\to 0$. Thus, we ask what happens if the latter
condition is violated. Here, we need to impose some weak dependence
assumptions on the underlying time series. Also, we state the result for
$H=\mathcal{L}$. We were unable to establish result for a general functional
$H$.

######  Lemma 3.3.5.

Assume that $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ holds and
$r_{n}^{\gamma+1}w_{n}\to\infty$. Assume that $\boldsymbol{X}$ is either
$\ell$-dependent or $\beta$-mixing with the rates

$\displaystyle\sum_{j=1}^{r_{n}}j^{\gamma}\beta_{j}=o(r_{n}^{\gamma+1}w_{n}^{2})\;.$
(3.4)

Then

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma+2}w_{n}^{2}}\mathbb{E}\left[(t(N_{1})-t(1))^{\gamma}\mathbbm{1}{\left\\{A_{1}\right\\}}\right]=\lim_{n\to\infty}\frac{\mathbb{E}\left[\mathcal{L}_{1}^{\gamma}\mathbbm{1}{\left\\{A_{1}\right\\}}\right]}{r_{n}^{\gamma+2}w_{n}^{2}}=\frac{1}{(\gamma+1)(\gamma+2)}{\vartheta}^{2}\;.$
(3.5)

We accompany the above results with several remarks in Section 5.1.6.

##  4 Weak convergence of empirical cluster process(es)

Let $H\in\mathcal{H}(\gamma)$. The statistics

$\displaystyle\frac{1}{nw_{n}}\sum_{j=1}^{m_{n}}H(\mathbb{X}_{j})=\frac{1}{nw_{n}}\sum_{j=1}^{m_{n}}H(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})=:\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$

can be considered as a disjoint block pseudo-estimator of the cluster index
${\boldsymbol{\nu}}^{*}(H)$. Indeed, under fairly general conditions one can
establish

$\displaystyle\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\stackrel{{\scriptstyle\tiny\mathbb{P}}}{{\longrightarrow}}{\boldsymbol{\nu}}^{*}(H)\;.$
(4.1)

To obtain an estimator, set $k=nw_{n}$ and $u_{n}$ is replaced with the $k$th
largest order statistics of $|\boldsymbol{X}_{j}|$, $j=1,\ldots,n$. Then, the
consistency result still holds. See [9, Section 10.1].

###  4.1 Functionals dominated by the cluster length

####  4.1.1 Small blocks scenario

We assume $r_{n}^{\gamma+1}w_{n}\to 0$. Hence, (4.1) is expected. We then
consider the empirical cluster process defined in (1.3):

$\displaystyle\widetilde{\mathbb{G}}_{n}(H)=\sqrt{nw_{n}}\left\\{\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)-\mathbb{E}[\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)]\right\\}=\frac{1}{\sqrt{nw_{n}}}\sum_{j=1}^{m_{n}}\left\\{H(\mathbb{X}_{j})-\mathbb{E}[H(\mathbb{X}_{j})]\right\\}\;,$

with the scaled blocks $\mathbb{X}_{j}$ defined in (1.1). To obtain its weak
convergence, we need to impose a stronger condition on the block size. We will
assume that $\mathcal{S}^{(\gamma(2+\delta))}(r_{n},u_{n})$ holds with an
arbitrary small $\delta>0$, which in turn implies, in particular, that (2.12a)
holds.

The following result extends Theorems 10.1.2 and 10.2.1 in [9] to unbounded
functionals. We state this (and the next results) with all the assumptions, to
be compatible with [9].

######  Theorem 4.1.1.

Let $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ be a stationary, regularly
varying $\mathbb{R}^{d}$-valued time series. Assume that
$\mathcal{R}(r_{n},u_{n})$ and $\beta(r_{n},\ell_{n})$ hold. Fix $\gamma>0$
and assume that there exists $\delta>0$ such that
$\mathcal{S}^{(\gamma(2+\delta))}(r_{n},u_{n})$ holds. Then:

  * •

For any $H\in\mathcal{H}(\gamma)$,
$\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$ is a consistent pseudo-
estimator of ${\boldsymbol{\nu}}^{*}(H)$;

  * •

$\widetilde{\mathbb{G}}_{n}\stackrel{{\scriptstyle\mbox{\tiny
fi.di.}}}{{\longrightarrow}}\widetilde{\mathbb{G}}$ on $\mathcal{H}(\gamma)$,
where $\widetilde{\mathbb{G}}$ is a centered Gaussian process with the
covariance functions ${\boldsymbol{\nu}}^{*}(HH^{\prime})$,
$H,H^{\prime}\in\mathcal{H}(\gamma)$.

We extend the above result to functional convergence. For this we need
additional assumptions on the function class. We consider
$\mathcal{G}\subseteq\mathcal{H}(\gamma)$ and equip it with the metric
$\rho^{*}(H,H^{\prime})={\boldsymbol{\nu}}^{*}(\\{H-H^{\prime}\\}^{2})$.
Define the random semi-metric $d_{n}$ on $\mathcal{G}$ by

$\displaystyle
d_{n}^{2}(H,H^{\prime})=\frac{1}{nw_{n}}\sum_{j=1}^{m_{n}}\\{H\left(\mathbb{X}_{j}\right)-H^{\prime}\left(\mathbb{X}_{j}\right)\\}\;.$

In what follows, we need a random covering number of the function class. In
the context of the stochastic processes considered here, we refer to Appendix
C in [9], while the general theory can be found in [17]. For statistical
applications it suffices to consider linearly ordered function classes of the
form $\mathcal{G}=\\{H_{s},s\in[s_{0},t_{0}]\\}$ for $0<s_{0}<t_{0}<\infty$
and some $H\in\mathcal{H}(\gamma)$. For such classes Conditions (b), (c), (d)
below are automatically fulfilled. See e.g. the proof of Theorem 10.3.1 in
[9].

The following result is a re-statement of Theorem C.4.5 in [9], adapted to the
setting of the present paper. See also Theorems 2.8 and 2.10 in [6]. A proof
is not provided.

######  Theorem 4.1.2.

Assume the conditions of Theorem 4.1.1 are satisfied. Assume also that:

  * (a)

The semi-metric space $(\mathcal{G},\rho)$ is totally bounded.

  * (b)

The envelope function
$\boldsymbol{G}(\boldsymbol{x}):=\sup_{H\in\mathcal{G}}H(\boldsymbol{x})$
belongs to $\mathcal{G}$.

  * (c)

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

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

  * (d)

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

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

The the process $\\{\mathbb{G}_{n}(H),H\in\mathcal{G}\\}$ is asymptotically
$\rho$-equicontinuous.

####  4.1.2 Moderate blocks scenario

We assume here that (2.12b) holds. Hence, the consistency in (4.1) is still
expected.

Let $\mathbb{X}_{j}^{\dagger}$, $j=1,\ldots,n$, be independent copies of
$\mathbb{X}_{j}$ and define

$\displaystyle\widetilde{\mathbb{G}_{n}}^{\dagger}(H):=\frac{1}{\sqrt{nw_{n}}}\sum_{j=1}^{m_{n}}\left\\{H(\mathbb{X}_{j}^{\dagger})-\mathbb{E}[H(\mathbb{X}_{j})]\right\\}\;.$

Thanks to the mixing assumption, the limiting behaviour of both
$\widetilde{\mathbb{G}_{n}}^{\dagger}$ and $\widetilde{\mathbb{G}_{n}}$
coincide. Set $H=\mathcal{L}^{\gamma}$. Then

$\displaystyle\mathrm{Var}\left(\widetilde{\mathbb{G}_{n}}^{\dagger}(H)\right)=\frac{\mathrm{Var}(H(\mathbb{X}_{1}))}{r_{n}w_{n}}=\underbrace{\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{2\gamma+2}w_{n}^{2}}}_{=:J_{n,1}}\left(r_{n}^{2\gamma+1}w_{n}\right)-\left(\underbrace{\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}}_{=:{J_{n,2}}}\right)^{2}r_{n}w_{n}\;.$

On account of Lemma 3.3.5 and Corollary 3.3.4, $J_{n,1}$ and $J_{n,2}$
converge to finite limits. (To be precise, convergence of $J_{n,2}$ requires
the anticlustering condition, which is almost equivalent to
$r_{n}^{\gamma+1}w_{n}\to 0$). Thus, under the moderate blocks condition the
variance of the of the empirical cluster process diverges to infinity.

Hence, the empirical process has to be modified. Define

$\displaystyle\widetilde{\mathbb{K}_{n}}(H):=\sqrt{\frac{n}{r_{n}^{2\gamma+1}}}\left\\{\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)-\mathbb{E}[\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)]\right\\}=\frac{1}{\sqrt{nr_{n}^{2\gamma+1}}w_{n}}\sum_{j=1}^{m_{n}}\left\\{H(\mathbb{X}_{j})-\mathbb{E}[H(\mathbb{X}_{j})]\right\\}\;.$

Then, with the obvious notation for the independent blocks process,

$\displaystyle\mathrm{Var}\left(\widetilde{\mathbb{K}_{n}}^{\dagger}(H)\right)=\frac{\mathrm{Var}(H(\mathbb{X}_{1}))}{r_{n}^{2\gamma+2}w_{n}^{2}}=\underbrace{\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{2\gamma+2}w_{n}^{2}}}_{=J_{n,1}}-\left(\underbrace{\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}}_{={J_{n,2}}}\right)^{2}r_{n}^{-2\gamma}\;.$

Since $\gamma>0$ and $r_{n}\to\infty$,
$\mathrm{Var}\left(\widetilde{\mathbb{K}_{n}}^{\dagger}(H)\right)$ converges
as $n\to\infty$. We make this analysis precise in the following limiting
statement.

######  Theorem 4.1.3.

Let $\\{\boldsymbol{X}_{j},j\in\mathbb{Z}\\}$ be a stationary, regularly
varying $\mathbb{R}^{d}$-valued time series. Fix $\gamma>0$. Assume that:

  * (i)

$\mathcal{R}(r_{n},u_{n})$ holds.

  * (ii)

$\boldsymbol{X}$ is either $\ell$-dependent or mixing with the rates
$\beta(r_{n},\ell_{n})$ and

$\displaystyle\sum_{j=1}^{r_{n}}j^{(2+\delta)\gamma}\beta_{j}=o(r_{n}^{(2+\delta)\gamma+1}w_{n}^{2})\;.$

for some $\delta>0$.

  * (iii)

$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ holds (hence $r_{n}^{\gamma+1}w_{n}\to
0$), while $r_{n}^{2\gamma+1}w_{n}\to\infty$.

  * (iv)

$nr_{n}w_{n}^{2}\to\infty$ and $r_{n}^{2\gamma+1}/n\to 0$ are satisfied.

Then:

  * •

For any $H\in\mathcal{H}(\gamma)$,
$\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)$ is a consistent pseudo-
estimator of ${\boldsymbol{\nu}}^{*}(H)$;

  * •

$\widetilde{\mathbb{K}}_{n}(\mathcal{L}^{\gamma})\stackrel{{\scriptstyle\mbox{\tiny\rm
d}}}{{\longrightarrow}}\widetilde{\mathbb{K}}(\mathcal{L}^{\gamma})$, where
$\widetilde{\mathbb{K}}(\mathcal{L}^{\gamma})$ is a centered normal random
variable with the variance $\\{(2\gamma+1)(2\gamma+2)\\}^{-1}{\vartheta}^{2}$.

####  4.1.3 Large blocks scenario

We finally assume that (2.12c) holds. In view of Corollary 3.3.4, the minimal
condition required for consistency is that
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ is satisfied, which in turn yields
$r_{n}^{\gamma+1}w_{n}\to 0$. If $r_{n}^{\gamma+1}w_{n}\to\infty$, then in
view of Lemma 3.3.5 one can only expect that

$\displaystyle\frac{1}{nr_{n}^{\gamma+1}w_{n}^{2}}\sum_{j=1}^{m_{n}}\mathcal{L}^{\gamma}(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})$

converges in probability to the right hand side of (3.5), which is not equal
to ${\boldsymbol{\nu}}^{*}(\mathcal{L}^{\gamma})$. In particular,
$\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\stackrel{{\scriptstyle\tiny\mathbb{P}}}{{\longrightarrow}}\infty$.

###  4.2 Location of jumps

Let $G$ be bounded as in Lemma 3.2.2. Let $H=T_{\rm max}^{\gamma}G$, so that
$H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})=t(N_{1})G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})$,
whenever there is a large jump in the block. In view of Lemma 3.2.2, the
consistency result (4.1) does not hold. The empirical cluster process has to
be re-defined again. Let

$\displaystyle\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H):=\frac{1}{nr_{n}^{\gamma}w_{n}}\sum_{j=1}^{m_{n}}H(u_{n}^{-1}\boldsymbol{X}_{(j-1)r_{n}+1,jr_{n}})\;.$

Thanks to Lemma 3.2.2,
$\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H)$ may serve as a
consistent estimator of $(\gamma+1)^{-1}{\boldsymbol{\nu}}^{*}(G)$. In this
spirit, we define another empirical cluster process:

$\displaystyle\widetilde{\mathbb{L}}_{n}(H)=\sqrt{nw_{n}}\left\\{\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H)-\mathbb{E}\left[\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H)\right]\right\\}=\frac{1}{\sqrt{nw_{n}}r_{n}^{\gamma}}\sum_{j=1}^{m_{n}}\left\\{H(\mathbb{X}_{j})-\mathbb{E}[H(\mathbb{X}_{j})]\right\\}\;.$

In other words,
$\widetilde{\mathbb{L}}_{n}(H)=\widetilde{\mathbb{G}}_{n}(H_{n})$, where
$H_{n}=H/r_{n}^{\gamma}$.

Then, with the obvious notation,

$\displaystyle\mathrm{Var}\left(\widetilde{\mathbb{L}_{n}}^{\dagger}(H)\right)=m_{n}\frac{\mathrm{Var}(H(\mathbb{X}_{1}))}{nr_{n}^{2\gamma}w_{n}}=\underbrace{\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{2\gamma+1}w_{n}}}_{=L_{n,1}}-\left(\underbrace{\frac{\mathbb{E}[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{\gamma+1}w_{n}}}_{={L_{n,2}}}\right)^{2}r_{n}w_{n}\;.$

By Lemma 3.2.2, the term $L_{n,1}$ converges to
$(2\gamma+1)^{-1}{\boldsymbol{\nu}}^{*}(G^{2})$. Also, $L_{n,2}$ converges and
hence the last term in the variance expansion vanishes. This motivates the
following theorem.

######  Theorem 4.2.1.

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

  * •

For any $G\in\mathcal{H}(0)$,
$\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(T_{\rm
max}^{\gamma}G)$ is a consistent pseudo-estimator of
$(1+\gamma)^{-1}{\boldsymbol{\nu}}^{*}(G)$;

  * •

$\widetilde{\mathbb{L}}_{n}\stackrel{{\scriptstyle\mbox{\tiny
fi.di.}}}{{\longrightarrow}}\widetilde{\mathbb{L}}$ on $\mathcal{H}(0)$, where
$\widetilde{\mathbb{L}}$ is a centered Gaussian process with the covariance
function $(1+\gamma)^{-1}{\boldsymbol{\nu}}^{*}(GG^{\prime})$,
$G,G^{\prime}\in\mathcal{H}(0)$.

######  Example 4.2.2.

Let $G(\boldsymbol{x})=\mathbbm{1}{\left\\{\boldsymbol{x}^{*}>1\right\\}}$. In
view of (5.1) below,

$\displaystyle\widetilde{\theta}_{n}:=(\gamma+1)\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(T_{\rm
max}^{\gamma}G)\wedge 1$

is a consistent pseudo-estimator the candidate extremal index ${\vartheta}$.
This type of estimator can help with a finite sample downward bias - the
classical blocks estimators are typically heavily biased towards zero. At the
same time, $\widetilde{\theta}_{n}$ has typically a higher variance as
compared to the classical blocks estimators.

##  5 Proofs I - unbounded cluster functionals

###  5.1 Consequences of the anticlustering conditions

####  5.1.1 Formulas for the candidate extremal index

Recall the notation
$\boldsymbol{x}^{*}=\sup_{j\in\mathbb{Z}}|\boldsymbol{x}_{j}|$. If
$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\boldsymbol{x}^{\ast}>1\right\\}}$,
since $|\boldsymbol{Y}_{0}|>1$, (2.5) gives

$\displaystyle{\boldsymbol{\nu}}^{*}(\mathbbm{1}{\left\\{\boldsymbol{x}^{\ast}>1\right\\}})=\mathbb{P}({\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1},{\boldsymbol{Y}^{\ast}>1})=\mathbb{P}({\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1})=\mathbb{P}({\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant 1})={\vartheta}\;.$
(5.1)

####  5.1.2 First and last jump decompositions

The following decompositions will play a crucial role:

$\displaystyle\mathbbm{1}{\left\\{\boldsymbol{X}_{1,r_{n}}^{\ast}>u_{n}\right\\}}=\sum_{i=1}^{r_{n}}\mathbbm{1}{\left\\{\boldsymbol{X}_{1,i-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{i}\right|>u_{n}\right\\}}\;.$

Recall the notation (2.10) for the exceedance times. Set $t(i)=t_{1}{(i)}$,
$i=1,\ldots,N_{1}$. Then

$\displaystyle\\{t(1)=j\\}=\\{\boldsymbol{X}_{1,j-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n}\\}\;,$ (5.2)

(with the convention $\boldsymbol{X}_{1,0}^{\ast}\equiv 0$). Likewise,

$\displaystyle\\{t(N_{1})=j\\}=\\{\boldsymbol{X}_{j+1,r_{n}}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n}\\}\;.$ (5.3)

(with the convention $\boldsymbol{X}_{r_{n}+1,r_{n}}^{\ast}\equiv 0$). The
above identities will be referred to as the first jump and the last jump
decomposition, respectively.

####  5.1.3 Conditional convergence of clusters

As a consequence of Proposition 2.7.1, for any bounded functional $H$ on
$\ell_{0}(\mathbb{R}^{d})$ that is almost surely continuous with respect to
the distribution of the tail process $\boldsymbol{Y}$ any integers
$i_{1},i_{2},j_{1},j_{2}$ and arbitrary $s\geqslant 0$,

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[H(u_{n}^{-1}(\boldsymbol{X}_{-[r_{n}s]-j_{1},-i_{1}},\boldsymbol{X}_{i_{2},j_{2}+[r_{n}s]}))\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$
$\displaystyle=\mathbb{E}\left[H(\boldsymbol{Y}_{-\infty,-i_{1}},\boldsymbol{Y}_{i_{2},\infty})\right]\;.$
(5.4)

This type of convergence will be referred to as the conditional convergence of
clusters.

####  5.1.4 Vague convergence of clusters

In view of the anticlustering condition $\mathcal{A}\mathcal{C}(r_{n},u_{n})$,
the following argument will be used repeatedly (cf. Proposition 2.8.1 and the
proof of Theorem 6.2.5 in [9]). Let $G$ be a bounded cluster functional such
that $G(\boldsymbol{x})=0$ whenever $\boldsymbol{x}^{\ast}\leqslant 1$. Then

$\displaystyle\frac{1}{r_{n}}\sum_{i=1}^{r_{n}}\mathbb{E}\left[G(u_{n}^{-1}\boldsymbol{X}_{1-i,r_{n}-i})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n}\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]=\int_{0}^{1}g_{n}(s)\mathrm{d}s$
(5.5)

with

$g_{n}(s)=\mathbb{E}\left[G(u_{n}^{-1}\boldsymbol{X}_{1-[r_{n}s],r_{n}-[r_{n}s]})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-[r_{n}s],-1}^{\ast}\leqslant
u_{n}\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]$

converging to
$g(s):=\mathbb{E}[G(\boldsymbol{Y})\mathbbm{1}{\left\\{\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1\right\\}}]$ for each $s\in(0,1)$. By the dominated convergence the limit of
the expression in (5.5) becomes

$\displaystyle\mathbb{E}[G(\boldsymbol{Y})\mathbbm{1}{\left\\{\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1\right\\}}]={\boldsymbol{\nu}}^{*}(G)\;.$

The first application is related to the candidate extremal index, cf. (2.7).
Next, we extend (5.5) in several directions. For $s\in(0,1)$ and integers
$i_{1},i_{2}\geqslant 1$, $0\leqslant j<i_{2}$, define

$\displaystyle g_{n}^{\rm two-
sided}(s;i_{1},j,i_{2}):=\mathbb{P}(\boldsymbol{X}_{1-[r_{n}s],-i_{1}}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{i_{2},r_{n}-[r_{n}s]}^{*}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\;.$

Then, (5.4) gives

$\displaystyle\lim_{n\to\infty}g_{n}^{\rm two-
sided}(s;i_{1},j,i_{2})=\mathbb{P}(\boldsymbol{Y}_{-\infty,-i_{1}}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{i_{2},\infty}^{\ast}\leqslant
1)\;$ (5.6)

for all $s\in(0,1)$. Likewise, let $G$ be a bounded cluster functional such
that $G(\boldsymbol{x})=0$ whenever $\boldsymbol{x}^{\ast}\leqslant 1$. Define

$\displaystyle g_{n}^{\rm left}(s;i_{1},G)$
$\displaystyle:=\mathbb{E}\left[G(u_{n}^{-1}\boldsymbol{X}_{1-[r_{n}s],r_{n}-[r_{n}s]})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-[r_{n}s],-i_{1}}^{\ast}\leqslant
u_{n}\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]\;,$ (5.7a)
$\displaystyle g_{n}^{\rm right}(s;i_{2},G)$
$\displaystyle:=\mathbb{E}\left[G(u_{n}^{-1}\boldsymbol{X}_{1-[r_{n}s],r_{n}-[r_{n}s]})\mathbbm{1}{\left\\{\boldsymbol{X}_{i_{2},r_{n}-[r_{n}s]}^{\ast}\leqslant
u_{n}\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]\;.$ (5.7b)

Then for all $s\in(0,1)$,

$\displaystyle\lim_{n\to\infty}g_{n}^{\rm left}(s;i_{1},G)$
$\displaystyle=\mathbb{E}[G(\boldsymbol{Y})\mathbbm{1}{\left\\{\boldsymbol{Y}_{-\infty,-i_{1}}^{\ast}\leqslant
1\right\\}}]\;,$ $\displaystyle\lim_{n\to\infty}g_{n}^{\rm right}(s;i_{2},G)$
$\displaystyle=\mathbb{E}[G(\boldsymbol{Y})\mathbbm{1}{\left\\{\boldsymbol{Y}_{i_{2},\infty}^{\ast}\leqslant
1\right\\}}]\;.$

In particular, if $i_{1}=i_{2}=1$, then

$\displaystyle\lim_{n\to\infty}g_{n}^{\rm
left}(s;1,G)=\lim_{n\to\infty}g_{n}^{\rm
right}(s;1,G)={\boldsymbol{\nu}}^{*}(G)\;.$ (5.9)

If moreover $G\equiv 1$, then we will simplify the notation $g_{n}^{\rm
left}(s;1,G)$ as $g_{n}^{\rm left}(s)$ and

$\displaystyle\lim_{n\to\infty}g_{n}^{\rm
left}(s)=\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1)={\vartheta}\;,\ \ \lim_{n\to\infty}g_{n}^{\rm
right}(s)=\mathbb{P}(\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant
1)={\vartheta}\;.$ (5.10)

A function $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is $\gamma$-homogeneous if for
all $a,s>0$, $f(as)=a^{\gamma}f(s)$. Denote

$\displaystyle f_{n}(s)=f\left(\frac{[r_{n}s]}{r_{n}}\right)\;.$

Since a homogeneous function is continuous, we have
$\lim_{n\to\infty}f_{n}(s)=f(s)$. Then, using (5.6) and the bounded
convergence,

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma+1}}\sum_{i=1}^{r_{n}}f(i)\mathbb{P}(\boldsymbol{X}_{1-i,-j_{1}}^{\ast}\leqslant
u_{n},\boldsymbol{X}_{j_{2},r_{n}-i}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}f_{n}(s)g_{n}^{\rm two-
sided}(s;j_{1},0,j_{2})\mathrm{d}s=\mathbb{P}(\boldsymbol{Y}_{-\infty,-j_{1}}^{\ast}\leqslant
1,\boldsymbol{Y}_{j_{2},\infty}^{\ast}\leqslant 1)\int_{0}^{1}f(s)\mathrm{d}s$
$\displaystyle=\frac{f(1)}{\gamma+1}\mathbb{P}(\boldsymbol{Y}_{-\infty,-j_{1}}^{\ast}\leqslant
1,\boldsymbol{Y}_{j_{2},\infty}^{\ast}\leqslant 1)\;,$ (5.11)

Let now $G$ be a bounded cluster functional such that $G(\boldsymbol{x})=0$
whenever $\boldsymbol{x}^{\ast}\leqslant 1$. Then, using (5.9),

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma+1}}\sum_{i=1}^{r_{n}}f(i)\mathbb{E}\left[G(\boldsymbol{X}_{1-i,r_{n}-i})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right\\}}\right]$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}f_{n}(s)g_{n}^{\rm
left}(s;1,G)\mathrm{d}s={\boldsymbol{\nu}}^{*}(G)\int_{0}^{1}f(s)\mathrm{d}s=\frac{f(1)}{\gamma+1}{\boldsymbol{\nu}}^{*}(G)\;,$
(5.12a)
$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma+1}}\sum_{i=1}^{r_{n}}f(i)\mathbb{E}\left[G(\boldsymbol{X}_{1-i,r_{n}-i})\mathbbm{1}{\left\\{\boldsymbol{X}_{1,r_{n}-i}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right\\}}\right]$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}f_{n}(s)g_{n}^{\rm
right}(s;1,G)\mathrm{d}s={\boldsymbol{\nu}}^{*}(G)\int_{0}^{1}f(s)\mathrm{d}s=\frac{f(1)}{\gamma+1}{\boldsymbol{\nu}}^{*}(G)\;.$
(5.12b)

Next, we will say that a function $f:\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$ is a
$(\gamma_{1},\gamma_{2})$-homogenous (with $\gamma_{1},\gamma_{2}\geqslant 0$)
if for all $a,b,s,t>0$, $f(as,bt)=a^{\gamma_{1}}b^{\gamma_{2}}f(s,t)$. We will
use the notation

$\displaystyle
f_{n}(s,t)=f\left(\frac{[r_{n}s]}{r_{n}},\frac{[r_{n}t]}{r_{n}}\right)\;.$

Note that $\lim_{n\to\infty}f_{n}(s,t)=f(s,t)$. As in (5.11),

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma_{1}+\gamma_{2}+1}}\sum_{i=1}^{r_{n}}f(i,i)\mathbb{P}(\boldsymbol{X}_{1-i,-j_{1}}^{\ast}\leqslant
u_{n},\boldsymbol{X}_{j_{2},r_{n}-i}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}f_{n}(s,s)g_{n}^{\rm two-
sided}(s;j_{1},0,j_{2})\mathrm{d}s=\mathbb{P}(\boldsymbol{Y}_{-\infty,-j_{1}}^{\ast}\leqslant
1,\boldsymbol{Y}_{j_{2},\infty}^{\ast}\leqslant
1)\int_{0}^{1}f(s,s)\mathrm{d}s$
$\displaystyle=\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}\mathbb{P}(\boldsymbol{Y}_{-\infty,-j_{1}}^{\ast}\leqslant
1,\boldsymbol{Y}_{j_{2},\infty}^{\ast}\leqslant 1)\;.$ (5.13)

Using (5.10), a similar argument gives

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma_{1}+\gamma_{2}+2}}\sum_{i_{1}=1}^{r_{n}}\sum_{i_{2}=i_{1}+1}^{r_{n}}f(i_{1},i_{2})\mathbb{P}(\boldsymbol{X}_{1-i_{1},-1}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\mathbb{P}(\boldsymbol{X}_{1,r_{n}-i_{2}}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}\left(\int_{s}^{1}f_{n}(s,t)g_{n}^{\rm
right}(t)\mathrm{d}t\right)g_{n}^{\rm
left}(s)\mathrm{d}s={\vartheta}^{2}f(1,1)\int_{0}^{1}s^{\gamma_{1}}\left(\int_{s}^{1}t^{\gamma_{2}}\mathrm{d}t\right)\mathrm{d}s\;.$
(5.14)

We will refer to this type of convergence as the vague convergence of
clusters.

####  5.1.5 Proofs for unbounded cluster functionals

###### Proof of Lemma 3.2.1.

The first jump decomposition in (5.2), stationarity, vague convergence of
clusters and the formula for ${\vartheta}$ in (5.1) give

$\displaystyle\frac{1}{r_{n}w_{n}}\mathbb{P}(\\{t(1)>qr_{n}\\}\cap
A_{1})=\frac{1}{r_{n}w_{n}}\sum_{i=qr_{n}+1}^{r_{n}}\mathbb{P}(t(1)=i,\boldsymbol{X}_{1,r_{n}}^{\ast}>u_{n})$
$\displaystyle=\frac{1}{r_{n}w_{n}}\sum_{i=qr_{n}+1}^{r_{n}}\mathbb{P}(\boldsymbol{X}_{1,i-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{i}\right|>u_{n})=\frac{1}{r_{n}}\sum_{i=qr_{n}+1}^{r_{n}}\mathbb{P}(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle\to(1-q)\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1)=(1-q){\vartheta}\;.$

The proof for $t(N_{1})$ is analogous. We use the last jump decomposition
(5.3) instead. ∎

###### Proof of Lemma 3.2.2.

We use the first jump decomposition:

$\displaystyle\mathbb{E}[f(t(1))G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]=\mathbb{E}[f(t(1))G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{A_{1}\right\\}}]$
$\displaystyle=w_{n}\sum_{i=1}^{r_{n}}f(i)\mathbb{E}\left[G(u_{n}^{-1}\boldsymbol{X}_{1-i,r_{n}-i})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n}\right\\}}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right]\;.$

Application of (5.12a) yields the result. The second part of the lemma follows
from the decomposition on the last jump and (5.12b). ∎

###### Proof of Lemma 3.2.4.

We use the first and the last jump decomposition simultaneously:

$\displaystyle\mathbb{E}[f(t(1),t(N_{1}))\mathbbm{1}{\left\\{A_{1}\right\\}}]$
$\displaystyle=\sum_{i_{1}=1}^{r_{n}}\sum_{i_{2}=i_{1}}^{r_{n}}f(i_{1},i_{2})\mathbb{P}(\boldsymbol{X}_{1,i_{1}-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{i_{1}}\right|>u_{n},\left|\boldsymbol{X}_{i_{2}}\right|>u_{n},\boldsymbol{X}_{i_{2}+1,r_{n}}^{\ast}\leqslant
u_{n})$
$\displaystyle=w_{n}\sum_{i=1}^{r_{n}}f(i,i)\mathbb{P}(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\boldsymbol{X}_{1,r_{n}-i}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})$
$\displaystyle\phantom{=}+\sum_{i=1}^{r_{n}}\sum_{j=1}^{\ell}f(i,j+i)\mathbb{P}(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-i}^{\ast}\leqslant
u_{n})$
$\displaystyle\phantom{=}+\sum_{i_{1}=1}^{r_{n}}\sum_{i_{2}=i_{1}+\ell+1}^{r_{n}}f(i_{1},i_{2})\mathbb{P}(\boldsymbol{X}_{1,i_{1}-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{i_{1}}\right|>u_{n},\left|\boldsymbol{X}_{i_{2}}\right|>u_{n},\boldsymbol{X}_{i_{2}+1,r_{n}}^{\ast}\leqslant
u_{n})$
$\displaystyle=:J_{1}(r_{n})+J_{2}(r_{n};\ell)+\widetilde{J}_{2}(r_{n};\ell)\;.$

Using (5.13) we have

$\displaystyle\lim_{n\to\infty}\frac{J_{1}(r_{n})}{r_{n}^{\gamma_{1}+\gamma_{2}+1}w_{n}}=\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant 1)\;.$ (5.15)

Set

$\displaystyle
D(r_{n};j)=\sum_{i=1}^{r_{n}}f(i,j+i)\mathbb{P}(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-i}^{\ast}\leqslant
u_{n})\;.$

Then

$\displaystyle J_{2}(r_{n};\ell)=\sum_{j=1}^{\ell}D(r_{n};j)\;,\ \
\widetilde{J}_{2}(r_{n};\ell)=\sum_{j=\ell+1}^{r_{n}}D(r_{n};j)\;.$

For each $j=1,\ldots,\ell$, using the convergence (5.6) we obtain

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma_{1}+\gamma_{2}+1}w_{n}}D(r_{n};j)=\lim_{n\to\infty}\int_{0}^{1}f_{n}\left(s,\frac{j}{r_{n}}+s\right)g_{n}^{\rm
two-sided}(s;1;j;j+1)\mathrm{d}s$
$\displaystyle=\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{j+1,\infty}^{\ast}\leqslant
1)\;.$

Hence,

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma_{1}+\gamma_{2}+1}w_{n}}J_{2}(r_{n};\ell)=\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}\sum_{j=1}^{\ell}\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{j+1,\infty}^{\ast}\leqslant
1)\;.$ (5.16)

Next,

$\displaystyle\widetilde{J}_{2}(r_{n};\ell)\leqslant\mathrm{cst}\sum_{i_{1}=1}^{r_{n}}\sum_{i_{2}=i_{1}+\ell+1}^{r_{n}}i_{1}^{\gamma_{1}}i_{2}^{\gamma_{2}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{i_{2}-i_{1}}\right|>u_{n})$
$\displaystyle=O(1)\sum_{i=1}^{r_{n}}i^{\gamma_{1}}\sum_{j=\ell+1}^{r_{n}-i}(j+i)^{\gamma_{2}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n})$
$\displaystyle=O(r_{n}^{\gamma_{1}+1}r_{n}^{\gamma_{2}})\sum_{j=\ell+1}^{r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n})$

and hence

$\displaystyle\frac{\widetilde{J}_{2}(r_{n};\ell)}{r_{n}^{\gamma_{1}+\gamma_{2}+1}w_{n}}=O(1)\frac{1}{w_{n}}\sum_{j=\ell+1}^{r_{n}}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n})$

and the latter term vanishes by letting $n\to\infty$ and then $\ell\to\infty$
on account of the stronger anticlustering condition
$\mathcal{S}^{(0)}(r_{n},u_{n})$.

Now, we combine (5.15)-(5.16) to get (keep in mind
$\left|\boldsymbol{Y}_{0}\right|>1$):’

$\displaystyle\frac{f(1,1)}{\gamma_{1}+\gamma_{2}+1}\sum_{j=0}^{\ell}\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{j+1,\infty}^{\ast}\leqslant
1)\;.$

Letting $\ell\to\infty$ gives

$\displaystyle\sum_{j=0}^{\infty}\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{j+1,\infty}^{\ast}\leqslant
1)=\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{0,\infty}^{\ast}>1)={\vartheta}\;.$

∎

###### Proof of Lemma 3.3.2.

The result follows directly from Proposition 2.8.1. Indeed,
$H(\boldsymbol{x})=\mathbbm{1}{\left\\{\mathcal{L}(\boldsymbol{x})>q\right\\}}$
is bounded, has support separated from ${\boldsymbol{0}}$ and is continuous
with respect to the law of $\boldsymbol{Y}$.

Moreover, $\mathcal{A}\mathcal{C}(r_{n},u_{n})$ gives
$\lim_{j\to\infty}|\boldsymbol{Y}_{j}|=0$ almost surely, hence

$\displaystyle\lim_{q\to\infty}\lim_{n\to\infty}\mathbb{P}\left(\mathcal{L}_{1}>q\mid
A_{1}\right)=0\;,$

that is the cluster length is tight under the conditional law. ∎

###### Proof of Lemma 3.3.3.

As in the proof of Lemma 3.2.4 we use the decompositions on the first and last
jumps simultaneously. Since $H\in\mathcal{H}(\gamma)$, we bound
$|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\leqslant C(t(N_{1})-t(1))^{\gamma}$.
Thus,

$\displaystyle\mathbb{E}\left[|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})>L\right\\}}\mathbbm{1}{\left\\{A_{1}\right\\}}\right]$
$\displaystyle\leqslant\mathrm{cst}\sum_{{i_{1}=1}\atop{i_{2}=i_{1}+L}}^{r_{n}}\mathbb{E}\left[|H|(u_{n}^{-1}\boldsymbol{X}_{i_{1},i_{2}})\mathbbm{1}{\left\\{\boldsymbol{X}_{1,i_{1}-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{i_{1}}\right|>u_{n},\left|\boldsymbol{X}_{i_{2}}\right|>u_{n},\boldsymbol{X}_{i_{2}+1,r_{n}}^{\ast}\leqslant
u_{n}\right\\}}\right]$
$\displaystyle=\mathrm{cst}\sum_{{i_{1}=1}\atop{i_{2}=i_{1}+L}}^{r_{n}}\mathbb{E}\left[|H|(u_{n}^{-1}\boldsymbol{X}_{0,i_{2}-i_{1}})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-i_{1},-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{i_{2}-i_{1}}\right|>u_{n},\boldsymbol{X}_{i_{2}-i_{1}+1,r_{n}-i_{1}}^{\ast}\leqslant
u_{n}\right\\}}\right]$
$\displaystyle=\mathrm{cst}\sum_{i=1}^{r_{n}}\sum_{j=L}^{r_{n}-i}\mathbb{E}\left[|H|(u_{n}^{-1}\boldsymbol{X}_{0,j})\mathbbm{1}{\left\\{\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-i}^{\ast}\leqslant
u_{n}\right\\}}\right]$ (5.17) $\displaystyle\leqslant
C\sum_{i=1}^{r_{n}}\sum_{j=L}^{r_{n}-i}j^{\gamma}\mathbb{P}(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-i}^{\ast}\leqslant
u_{n})$ $\displaystyle\leqslant
Cr_{n}\sum_{j=L}^{r_{n}}j^{\gamma}\mathbb{P}(\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n})\;.$

We conclude by applying $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$. ∎

###### Proof of Lemma 3.3.5.

We continue with (5.1.5) by setting $L=1$. Here, $H=\mathcal{L}$. Hence,

$\displaystyle\mathbb{E}\left[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{A_{1}\right\\}}\right]$
$\displaystyle=\sum_{j=1}^{r_{n}}j^{\gamma}\sum_{i=1}^{r_{n}-j}\mathbb{P}\left(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{0}\right|>u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-i}^{\ast}\leqslant
u_{n}\right)$
$\displaystyle=:\sum_{j=1}^{r_{n}}j^{\gamma}D(r_{n};j)=\sum_{j=1}^{\ell}j^{\gamma}D(r_{n};j)+\sum_{j=\ell+1}^{r_{n}}j^{\gamma}D(r_{n};j)$
$\displaystyle=:J(r_{n};\ell)+\widetilde{J}(r_{n};\ell)\;.$

We have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}D(r_{n};j)$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}\mathbb{P}(\boldsymbol{X}_{1-[r_{n}s],-1}^{\ast}\leqslant
u_{n},\left|\boldsymbol{X}_{j}\right|>u_{n},\boldsymbol{X}_{j+1,r_{n}-[r_{n}s]}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n})\mathrm{d}s$
$\displaystyle=\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{j}\right|>1,\boldsymbol{Y}_{j+1,\infty}^{\ast}\leqslant
1)\;.$

Hence, $J(r_{n};\ell)=O_{P}(r_{n}w_{n})$.

Next, in case of $\ell$-dependence,

$\displaystyle\widetilde{J}(r_{n};\ell)=w_{n}^{2}\sum_{j=\ell+1}^{r_{n}}j^{\gamma}\sum_{i=1}^{r_{n}-j}\mathbb{P}\left(\boldsymbol{X}_{1-i,-1}^{\ast}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right)\mathbb{P}\left(\boldsymbol{X}^{\ast}_{1,r_{n}-i-j}\leqslant
u_{n}\mid\left|\boldsymbol{X}_{0}\right|>u_{n}\right)$
$\displaystyle=:\widetilde{J}^{*}(r_{n};\ell)\;.$

The term $\widetilde{J}^{*}(r_{n};\ell)$ converges at the rate
$r_{n}^{\gamma+2}w_{n}^{2}$. Indeed, using the notation (5.7a)-(5.7b) and the
limit in (5.10), bearing in mind $[r_{n}s]+[r_{n}t]\sim[r_{n}(s+t)]$, we have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{\gamma+2}w_{n}^{2}}\mathbb{E}\left[H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{A_{1}\right\\}}\right]$
$\displaystyle=\lim_{n\to\infty}\int_{0}^{1}t^{\gamma}\left(\int_{0}^{1-t}g_{n}^{\rm
left}(s)g_{n}^{\rm
right}(s+t)\mathrm{d}s\right)\mathrm{d}t=\frac{{\vartheta}^{2}}{(\gamma+1)(\gamma+2)}\;.$

In case of $\beta$-mixing,

$\displaystyle\widetilde{J}(r_{n};\ell)=\widetilde{J}^{*}(r_{n};\ell)+J_{\beta}(r_{n})\;,$

where

$\displaystyle
J_{\beta}(r_{n})\leq\mathrm{cst}\sum_{j=\ell+1}^{r_{n}}j^{\gamma}\sum_{i=1}^{r_{n}-j}\beta_{j}=\mathrm{cst}\sum_{j=\ell+1}^{r_{n}}j^{\gamma}(r_{n}-j)\beta_{j}\leq\mathrm{cst}\
r_{n}\sum_{j=\ell+1}^{r_{n}}j^{\gamma}\beta_{j}\;.$

By the assumption, $J_{\beta}(r_{n})=o(r_{n}^{\gamma+2}w_{n}^{2})$. The proof
for the large blocks is concluded. ∎

####  5.1.6 Additional comments for unbounded cluster functionals

######  Remark 5.1.

The result of Lemma 3.3.5 should be compared to Lemma 3.2.4. Take
$\gamma=\gamma_{1}=\gamma_{2}=1$ and $f(s,t)=(t-s)_{+}$. Then Lemma 3.2.4
gives
$\mathbb{E}[(t(N_{1})-t(1))\mathbbm{1}{\left\\{A_{1}\right\\}}]/(r_{n}^{3}w_{n})\to
0$. Lemma 3.3.5 allows to recover non-zero limit when scaled by
$r_{n}^{3}w_{n}^{2}$: $t(1)$ and $t(N_{1})$ become independent when normalized
by a smaller value. Indeed, assume that $U$ and $V$ are independent, standard
uniform random variables. Then
$\mathbb{E}[(U-V)_{+}^{\gamma}]=1/((\gamma+1)(\gamma+2))$, which can be
recognized as the right hand side of (3.5).

######  Remark 5.2.

The mixing condition (3.4) is somehow restrictive. Assume that
$\beta_{j}=O(j^{-\delta})$, $\delta>0$. Then (3.4) holds whenever
$r_{n}^{\delta}w_{n}^{2}\to\infty$. Thus, $\delta$ has to be sufficiently
large. In particular, the condition holds if the mixing coefficients decay at
the exponential rate.

######  Remark 5.3.

The proof of tightness in Lemma 3.3.2 requires the anticlustering
$\mathcal{A}\mathcal{C}(r_{n},u_{n})$ only. The convergence of the moments
requires the stronger condition $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$. In
i.i.d. case, the stronger anticlustering condition is equivalent to
$r_{n}^{\gamma+1}w_{n}=o(1)$; cf. Remark 2.2. We argue that the latter
assumption is necessary for convergence of moments of $\mathcal{L}_{1}$.

Assume that $\boldsymbol{X}$ is i.i.d. Then, the cluster length of the tail
process is
$\mathcal{L}(\boldsymbol{Y})\stackrel{{\scriptstyle\text{d}}}{{=}}1$. We find
sufficient and necessary conditions so that

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\mathcal{L}_{1}\mid
A_{1}\right]=1\;.$ (5.18)

We have as $n\to\infty$,

$\displaystyle f(1)$ $\displaystyle:=\mathbb{P}\left(\mathcal{L}_{1}=1\mid
A_{1}\right)=\binom{r_{n}}{1}w_{n}(1-w_{n})^{r_{n}-1}\bigg{/}\Big{[}1-(1-w_{n})^{r_{n}}\Big{]}\sim
1.$

Next, for any $i\geqslant 2$, we have

$\displaystyle f(i)$ $\displaystyle:=\mathbb{P}\left(\mathcal{L}_{1}=i\mid
A_{1}\right)$
$\displaystyle=\frac{1}{1-(1-w_{n})^{r_{n}}}\sum_{j=1}^{r_{n}-i+1}\underbrace{(1-w_{n})^{j-1}w_{n}}_{t(1)=j}\cdot\underbrace{(1-w_{n})^{r_{n}-(j+i-1)}w_{n}}_{t(N_{1})=j+i-1}$
$\displaystyle=\frac{w_{n}^{2}(r_{n}-i+1)(1-w_{n})^{r_{n}-i}}{1-(1-w_{n})^{r_{n}}}\sim\frac{r_{n}-i+1}{r_{n}}w_{n}.$

uniformly for all $2\leqslant i\leqslant r_{n}$.

Clearly, the limit in (5.18) is equivalent to
$\sum_{i=2}^{r_{n}}i^{\gamma}f(i)\to 0$. It suffices to note that

$\displaystyle\sum_{i=2}^{r_{n}}i^{\gamma}f(i)$
$\displaystyle\sim\sum_{i=2}^{r_{n}}i^{\gamma}w_{n}\frac{r_{n}-i+1}{r_{n}}$
$\displaystyle=r_{n}^{\gamma+1}w_{n}\sum_{i=2}^{r_{n}}\left(\frac{i}{r_{n}}\right)^{\gamma}\cdot\frac{r_{n}-i+1}{r_{n}}\cdot\frac{1}{r_{n}}\sim
r_{n}^{\gamma+1}w_{n}{\rm Beta}(\gamma+1,2)\;,$

where ${\rm Beta}(\cdot,\cdot)$ is the standard Beta function. This yields
that the condition $r_{n}^{\gamma+1}w_{n}\to 0$ is necessary (and sufficient)
for the conditional convergence of moments of the cluster length.

######  Remark 5.4.

From (3.3) we obtain for $q\geqslant 1$,

$\displaystyle\lim_{n\to\infty}\mathbb{P}\left(\mathcal{L}_{1}=q\mid
A_{1}\right)=\frac{\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{q-1}\right|>1,\boldsymbol{Y}_{q,\infty}^{\ast}\leqslant
1)}{{\vartheta}}=:f(q)\;.$

Then, since $\left|\boldsymbol{Y}_{0}\right|>1$, bearing in mind (5.1),

$\displaystyle f(1)+\sum_{q=2}^{\infty}f(q)$
$\displaystyle=\frac{\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant
1)}{{\vartheta}}+\frac{1}{{\vartheta}}\sum_{q=2}^{\infty}{\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\left|\boldsymbol{Y}_{q-1}\right|>1,\boldsymbol{Y}_{q,\infty}^{\ast}\leqslant
1)}$
$\displaystyle=\frac{\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{1,\infty}^{\ast}\leqslant
1)}{{\vartheta}}+\frac{\mathbb{P}(\boldsymbol{Y}_{-\infty,-1}^{\ast}\leqslant
1,\boldsymbol{Y}_{1,\infty}^{\ast}>1)}{{\vartheta}}=1\;.$

Hence, the limiting conditional distribution of the cluster length is a proper
distribution.

######  Remark 5.5.

Take $q=0$. The the left hand side of (3.3) is by the definition of the
cluster length equal to 1. At the same time, since $|\boldsymbol{Y}_{0}|>1$,
using (5.1), the right hand side of (3.3) also reduces to one. In case of
extremal independence, $\boldsymbol{Y}_{1,\infty}^{\ast}\equiv 0$ and hence
$\lim_{n\to\infty}\mathbb{P}(\mathcal{L}_{1}>1\mid A_{1})=0$ as expected.

##  6 Proofs II - the empirical cluster processes

###### Proof of Theorem 4.1.1.

We start with consistency. We note that
$\mathcal{S}^{(\gamma(2+\delta))}(r_{n},u_{n})$ implies
$\mathcal{S}^{(\gamma)}(r_{n},u_{n})$. Hence Corollary 3.3.4 gives
$\lim_{n\to\infty}\mathbb{E}\left[\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\right]={\boldsymbol{\nu}}^{*}(H)$.
Thanks to the mixing assumption, we can treat the blocks as independent. Then

$\displaystyle\mathrm{Var}\left(\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\right)\leq
m_{n}\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{n^{2}w_{n}^{2}}=\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}\frac{1}{nw_{n}}\;.$

Since $\mathcal{S}^{(2\gamma)}(r_{n},u_{n})$ holds, by Corollary 3.3.4 and
$nw_{n}\to\infty$, the variance of the estimator vanishes. Consistency has
been established.

Next, we verify that the class $\mathcal{H}(\gamma)$ fulfills the conditions
of Theorem 10.2.1 in [9].

  * (BCLT1)

This conditions states that (1.2) holds. This in turn is guaranteed by Lemma
3.3.3.

  * (BCLT2)

We need to show that for any $\eta>0$ we have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}w_{n}}\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})>\eta\sqrt{nw_{n}}\right\\}}]=0\;.$

The left hand-side is bounded by

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[|H|^{2+\delta}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}w_{n}}\frac{1}{(nw_{n})^{\delta/2}}$

which vanishes on account of the anticlustering assumption and
$nw_{n}\to\infty$. Here, the anticlustering condition with $\gamma(2+\delta)$
is needed. Indeed, if $H\in\mathcal{H}(\gamma)$, then $|H|^{2+\delta}$ belongs
to $\mathcal{H}(\gamma(2+\delta))$.

  * (BCLT3)

We need to verify that there exist functions
$K_{n}:(\mathbb{R}^{d})^{\ell_{n}}\to\mathbb{R}$ such that

$\displaystyle\left|H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})-H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}-\ell_{n}})\right|\leq
K_{n}(\boldsymbol{X}_{r_{n}-\ell_{n}+1,r_{n}})$

and

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[K_{n}(\boldsymbol{X}_{r_{n}-\ell_{n}+1,r_{n}})]}{r_{n}w_{n}}=0\;.$

On account of Assumption 2.3.1(iii),
$H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\not=H(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}-\ell_{n}})$
if and only if $\boldsymbol{X}^{*}_{r_{n}-\ell_{n}+1,r_{n}}>u_{n}$. Thus,
choose
$K_{n}(\boldsymbol{X}_{r_{n}-\ell_{n}+1,r_{n}})=\mathbbm{1}{\left\\{\boldsymbol{X}^{*}_{r_{n}-\ell_{n}+1,r_{n}}>u_{n}\right\\}}$
and then

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

∎

###### Proof of Theorem 4.1.3.

The proof is almost identical to the one for Theorem 4.1.1.

Since $\mathcal{S}^{(\gamma)}(r_{n},u_{n})$ holds, Corollary 3.3.4 gives
$\lim_{n\to\infty}\mathbb{E}\left[\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\right]={\boldsymbol{\nu}}^{*}(H)$.
Thanks to the mixing assumption, we can treat the blocks as independent. Then

$\displaystyle\mathrm{Var}\left(\widetilde{\boldsymbol{\nu}}_{n,r_{n}}^{*}(H)\right)\leq
m_{n}\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{n^{2}w_{n}^{2}}=\frac{\mathbb{E}[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{2\gamma+2}w_{n}^{2}}\frac{r_{n}^{2\gamma+1}}{n}\;.$

Since $r_{n}^{2\gamma+1}w_{n}\to\infty$, we use Lemma 3.3.5 to conclude
convergence of the second last ratio. At the same time.
$r_{n}^{2\gamma+1}/n\to 0$ by the assumption. Consistency has been
established.

For the weak convergence, only the modified version of the Lindeberg condition
(BCLT2) has to be proven. We need to show that for any $\eta>0$ we have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{2\gamma+2}w_{n}^{2}}\mathbb{E}\left[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})>\eta\sqrt{nr_{n}^{2\gamma+1}}w_{n}\right\\}}\right]=0\;.$

The left hand-side is bounded by

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[|H|^{2+\delta}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{(2+\delta)\gamma+2}w_{n}^{2}}\frac{1}{(nr_{n}w_{n}^{2})^{\delta/2}}\;.$

The first ratio converges by Lemma 3.3.5 (thanks to the assumptions
(ii)-(iii)), while the second one vanishes by the assumption (iv). ∎

###### Proof of Theorem 4.2.1.

Thanks to Lemma 3.2.2,

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H)\right]=\lim_{n\to\infty}\frac{\mathbb{E}[T_{\rm
max}^{\gamma}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})G(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{\gamma+1}w_{n}}=\frac{1}{\gamma+1}{\boldsymbol{\nu}}^{*}(G)\;.$

Again, the blocks can be treated as independent and hence

$\displaystyle\mathrm{Var}\left(\widetilde{\widetilde{\boldsymbol{\nu}}}_{n,r_{n}}^{*}(H)\right)\leq\frac{\mathbb{E}[T_{\rm
max}^{2\gamma}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})G^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{2\gamma+1}w_{n}}\frac{1}{nw_{n}}\to
0\;.$

For the weak convergence, again, only the modified version of the Lindeberg
condition (BCLT2) has to be proven. We need to show that for any $\eta>0$ we
have

$\displaystyle\lim_{n\to\infty}\frac{1}{r_{n}^{2\gamma+1}w_{n}}\mathbb{E}\left[H^{2}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})\mathbbm{1}{\left\\{|H|(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})>\eta\sqrt{nr_{n}^{2\gamma}w_{n}}\right\\}}\right]=0\;.$

The left hand-side is bounded by

$\displaystyle\lim_{n\to\infty}\frac{\mathbb{E}[|H|^{2+\delta}(u_{n}^{-1}\boldsymbol{X}_{1,r_{n}})]}{r_{n}^{(2+\delta)\gamma+1}w_{n}}\frac{1}{(nw_{n})^{\delta/2}}\;$

and vanishes by Lemma 3.2.2 and $nw_{n}\to\infty$. ∎

## Acknowledgements

The authors are grateful to Philippe Soulier for an extensive discussion. Both
authors were supported by an NSERC grant. A part of this research was
conducted during the second author’s stay at the Sydney Mathematical Research
Institute (October - November 2022). Rafał Kulik thanks for the support and
hospitality provided by SMRI.

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