Gumbel and Fr\'echet convergence of the maxima of independent random walks

TL;DR

This paper develops asymptotic theory for the maxima of independent random walks, showing their convergence to Gumbel or Fréchet distributions, based on large deviation principles.

Contribution

It establishes the convergence of the maximum of iid random walks to extreme value distributions, extending existing results with precise large deviation techniques.

Findings

Maximum of iid random walks converges to Gumbel or Fréchet distributions.

Convergence results depend on large deviation principles for sums of independent variables.

Provides a rigorous foundation for extreme value analysis of random walks.

Abstract

We consider point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fr\'echet distributions. The proofs heavily depend on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.

Tables

Table 1. Table 1. Separating sequences ( ξ n ) subscript 𝜉 𝑛 (\xi_{n}) and ( ψ n ) subscript 𝜓 𝑛 (\psi_{n}) for the normal and subexponential approximations of ℙ ​ ( S n − 𝔼 ​ [ S n ] > x ) ℙ subscript 𝑆 𝑛 𝔼 delimited-[] subscript 𝑆 𝑛 𝑥 {\mathbb{P}}(S_{n}-\mathbb{E}[S_{n}]>x) . We also assume var ​ ( X ) = 1 var 𝑋 1 {\rm var}(X)=1 . Here ( h n ) , ( h ~ n ) subscript ℎ 𝑛 subscript ~ ℎ 𝑛 (h_{n}),(\widetilde{h}_{n}) are any sequences converging to infinity. For completeness, we also include the regularly varying class RV ​ ( α ) RV 𝛼 {\rm RV}(\alpha) . The table is taken from Mikosch and Nagaev  [ 19 ] .

$F\in$	${\xi }_n$	${\psi }_n$
$\mathrm{RV}\left(\alpha \right)$ , $\alpha >2$	${\left(\left(\alpha -2\right)n\log n\right)}^{1/2}$	${\left(\left(\alpha -2\right)n\log n\right)}^{1/2}$
$\mathrm{LN}\left(\gamma \right)$ ,	${\left(n{\left(\log n\right)}^{\gamma }\right)}^{1/2}$	${\left(n{\left(\log n\right)}^{\gamma }\right)}^{1/2}$
$\mathrm{LN}\left(\gamma \right)$ , $\gamma \ge 2$	${\left(n{\left(\log n\right)}^{\gamma }\right)}^{1/2}/{\tilde{h}}_n$	$n^{1/2}{\left(\log n\right)}^{\gamma -1}{h}_n$
$\mathrm{WE}\left(\tau \right)$ ,	$n^{1/\left(2-\tau \right)}/{\tilde{h}}_n$	$n^{1/\left(2-2\tau \right)}{h}_n$
$\mathrm{WE}\left(\tau \right)$ ,	$n^{2/3}/{\tilde{h}}_n$	$n^{1/\left(2-2\tau \right)}{h}_n$

Table 2. Table 2. Upper bounds for p 𝑝 p .

Example No	Upper bound for $p$
4.4 Petrov case	$\exp \left(o\left(n^{1/3}\right)\right)$
4.5 , $s>2$	$n^{\left(s-2\right)/2}$
4.6 $\mathrm{RV}\left(\alpha \right)$, $\alpha >2$,	$n^{c/2}$
4.7 $\mathrm{LN}\left(\gamma \right)$, $\gamma >1$	$\exp \left(o\left({\left(\log n\right)}^{\gamma }\right)\right)$
4.8 $\mathrm{WE}\left(\tau \right)$, $\tau \le 0.5$	$\exp \left(o\left(n^{\tau /\left(2-\tau \right)}\right)\right)$
4.8 $\mathrm{WE}\left(\tau \right)$, $\tau \in (0.5,1)$	$\exp \left(o\left(n^{1/3}\right)\right)$

Table 3. Table 3. Lower bounds on the block size r n subscript 𝑟 𝑛 r_{n}

Example No	Lower bounds for $\left(r_n\right)$
4.4 Petrov case	$r_n/{\left(\log n\right)}^3\to \infty$
4.6 $\mathrm{RV}\left(\alpha \right)$, $\alpha >2$	$r_n>n^{2/\left(\alpha -\epsilon \right)}$ any $\epsilon \in (0,\alpha -2)$
4.5 , $s>2$	$r_n>n^{2/s}$
4.7 $\mathrm{LN}\left(\gamma \right)$, $\gamma >1$	$r_n/\exp \left({\left(2\log n\right)}^{1/\gamma }\right)\to \infty$
4.8 $\mathrm{WE}\left(\tau \right)$, $\tau \le 0.5$	$r_n/{\left(\log n\right)}^{\left(2-\tau \right)/\tau }\to \infty$
4.8 $\mathrm{WE}\left(\tau \right)$, $\tau \in (0.5,1)$	$r_n/{\left(\log n\right)}^3\to \infty$

Equations

$$S_0=0,S_n=X_1+\cdots +X_n,n\ge 1.$$

$$N_p=\sum _{i=1}^p{\epsilon }_{c_n^{-1}(S_{ni}-d_n)}\stackrel{d}{\to }N,n\to \infty ,$$

$$\displaystyle p_{n}\,{\mathbb{P}}\big{(}c_{n}^{-1}(S_{n}-d_{n})\in(a,b]\big{)}\to\mu(a,b]\,,$$

$$\mathbb{P}(S_n>x)\sim n\overline{F}(x),x\to \infty .$$

$$F(-x)\sim p_{-}{x}^{-\alpha }L(x)\text{mbox}and\overline{F}(x)\sim p_{+}{x}^{-\alpha }L(x),x\to \infty .$$

$$c_n^{-1}(M_n-d_n)\stackrel{d}{\to }Y\sim H,n\to \infty .$$

$$\overline{F}(x)=\frac{L(x)}{x^{\alpha }},x>0;$$

$$c_n^{-1}{M}_n\stackrel{d}{\to }Y\sim {\Phi }_{\alpha },n\to \infty ,$$

$$\underset{u\to \infty }{\lim }\frac{\overline{F}(u+a(u)x)}{\overline{F}(u)}={\mathrm{e}}^{-x},x\in \mathbb{R};$$

$$c_n^{-1}(M_n-d_n)\stackrel{d}{\to }Y\sim \Lambda ,n\to \infty ,$$

$$d_n=\sqrt{2\log n}-\frac{\log \log n+\log 4\pi }{2(2\log n{)}^{1/2}}.$$

$$\displaystyle c_{n}=d_{n}/\sqrt{2\log n}\quad\mbox{and}\quad d_{n}=\exp\Big{(}\sqrt{2\log n}-\frac{\log\log n+\log 4\pi}{2(2\log n)^{1/2}}\Big{)}\,;$$

$$\displaystyle c_{n}=(\lambda\tau)^{-1}s_{n}^{1/\tau-1}\quad\mbox{and}\quad d_{n}=s_{n}^{1/\tau}+\dfrac{1}{\tau}s_{n}^{1/\tau-1}\Big{(}\dfrac{\beta}{\lambda\tau}\log s_{n}+\dfrac{\log c}{\lambda}\Big{)}\,,$$

$${\tilde{N}}_p=\sum _{i=1}^p{\epsilon }_{X_{ni}}\stackrel{d}{\to }N,n\to \infty ,$$

$$p\mathbb{P}(X_{n1}\in \cdot )\stackrel{v}{\to }\mu (\cdot ),n\to \infty ,$$

$$\displaystyle p\,{\mathbb{P}}\big{(}c_{n}^{-1}(S_{n}-d_{n})\in(a,b])\to\mu(a,b]\quad\mbox{or}\quad p\,{\mathbb{P}}\big{(}c_{n}^{-1}(S_{n}-d_{n})>a)\to\mu(a,\infty)\,,$$

$$\displaystyle\dfrac{{\mathbb{P}}(S_{n}/\sqrt{n}>x)}{\overline{\Phi}(x)}=\exp\Big{(}\dfrac{x^{3}}{\sqrt{n}}\lambda\big{(}\dfrac{x}{\sqrt{n}}\big{)}\Big{)}\Big{[}1+O\big{(}\dfrac{x+1}{\sqrt{n}}\big{)}\Big{]}\,,\quad n\to\infty\,.$$

$$\frac{\mathbb{P}(S_n/\sqrt{n}>x)}{\overline{\Phi }(x)}\to 1,n\to \infty .$$

$$\displaystyle\lim_{x\to\infty}\dfrac{\overline{F}(x+y)}{\overline{F}(x)}=1\,,\quad\mbox{for any real $y$,}$$

$$\underset{n\to \infty }{\lim }\underset{x>{\gamma }_n}{\sup }\frac{\mathbb{P}(S_n>x)}{n\overline{F}(x)}\le 1.$$

$$\displaystyle\lim_{n\to\infty}\sup_{x>\gamma_{n}}\Big{|}\dfrac{{\mathbb{P}}(S_{n}>x)}{n\,\overline{F}(x)}-1\Big{|}=0\,.$$

$$\mathbb{P}(S_n>x)=\overline{\Phi }(x/\sqrt{n})(1+o(1))+n\overline{F}(x)(1+o(1)).$$

$$\displaystyle\sup_{1<x/\sqrt{n}<\sqrt{c_{1}\,\log n}}\Big{|}\dfrac{{\mathbb{P}}(\pm S_{n}>x)}{\overline{\Phi}\big{(}x/\sqrt{n}\big{)}}-1\Big{|}\to 0\,,\qquad n\to\infty\,,$$

$$\displaystyle\sup_{x/\sqrt{n}>\sqrt{c_{2}\,\log n}}\Big{|}\dfrac{{\mathbb{P}}(\pm S_{n}>x)}{n\,{\mathbb{P}}(|X|>x)}-p_{\pm}\Big{|}\to 0\,,\qquad n\to\infty\,.$$

$$n\mathbb{P}(|X|>a_n)+\frac{n}{a_n^2}\mathbb{E}[X^{2}1(|X|\le a_n)]=1,n=1,2,\dots ,$$

$$\underset{n\to \infty }{\lim }\underset{x>{\gamma }_n}{\sup }\frac{n}{x^2}\frac{\mathbb{E}[X^{2}1(|X|\le x)]}{[n\mathbb{P}(|X|>x){]}^{\delta }}=0.$$

$$\displaystyle d_{n}=\left\{\begin{array}[]{ll}0\,,&\quad\alpha\in(0,1)\cup(1,2]\,,\\[5.69054pt] n\;{\mathbb{E}}[X\,{\mathbf{1}}(|X|\leq a_{n})]\,,&\quad\alpha=1\,.\end{array}\right.$$

$$\displaystyle\sup_{x>\gamma_{n}}\Big{|}\dfrac{{\mathbb{P}}\big{(}\pm(S_{n}-d_{n})>x\big{)}}{n\,{\mathbb{P}}(|X|>x)}-p_{\pm}\Big{|}\to 0\,,\qquad n\to\infty\,.$$

$$\displaystyle\sup_{x<\xi_{n}}\Big{|}\dfrac{{\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>x)}{\overline{\Phi}(x/\sqrt{n})}-1\Big{|}\to 0\,,$$

Full text

Thomas Mikosch’s research is partly support by an Alexander von Humboldt Research Award. He takes pleasure in thanking the Faculty of Mathematics of Ruhruniversität Bochum for hosting him in the period December 2018–May 2019.

1. Introduction

Let $(X_{i})$ be an iid sequence of random variables with generic element $X$, distribution $F$ and right tail $\overline{F}=1-F$. Define the corresponding partial sum process

[TABLE]

Consider iid copies $(S_{ni})_{i=1,2,\ldots}$ of $S_{n}$. We also introduce an integer sequence $(p_{n})$ such that $p=p_{n}\to\infty$ as $n\to\infty$. We are interested in the limiting behavior of the $k$ largest values among $(S_{ni})_{i=1,\ldots,p}$, in particular in the possible limit laws of the maximum $\max_{i=1,\ldots,p}S_{ni}$. More generally, writing $\varepsilon_{x}$ for Dirac measure at $x$, we are interested in the limiting behavior of the point processes

[TABLE]

for suitable constants $c_{n}>0$ and $d_{n}\in{\mathbb{R}}$ toward a Poisson random measure $N$ with Radon mean measure $\mu$ (we write ${\rm PRM}(\mu)$).

Our main motivation for this work comes from random matrix theory, in particular when dealing with sample covariance matrices. Their entries are dependent random walks. However, in various situations the theory can be modified in such a way that it suffices to study independent random walks. We refer to Section 4.6 for a discussion.

Relation (1.1) is equivalent to the following limit relations for the tails

[TABLE]

for any $a<b$ provided that $\mu(a,b]<\infty$; see Resnick [29], Theorem 5.3. These conditions involve precise large deviation probabilities for the random walk $(S_{n})$; in Section 3 we provide some results which are relevant in this context.

We distinguish between two types of precise large deviation results:

• •

normal approximation

• •

subexponential approximation

The normal approximation can be understood as extension of the central limit theorem for $(S_{n}/\sqrt{n})$ toward increasing intervals. This approximation causes the maxima of $(S_{ni}/\sqrt{n})$ to behave like the maxima of an iid normal sequence, i.e., these maxima converge in distribution to the Gumbel distribution. This is in contrast to the subexponential approximation which requires that $F$ is a so-called subexponential distribution; see Section 2.1. In particular, $\overline{F}$ is heavy-tailed in the sense that the moment generating function does not exist. This fact implies that ${\mathbb{P}}(S_{n}>x_{n})\sim n\,\overline{F}(x_{n})$ for sufficiently fast increasing sequences $x_{n}\to\infty$. Hence $n\,\overline{F}(x_{n})$ dominates ${\mathbb{P}}(S_{n}>x_{n})$ at sufficiently high levels $x_{n}$ and, as in limit theory for the maxima of an iid sequence, $\overline{F}$ determines the type of the limit distribution of the maxima of $(S_{ni})$ as well as the normalizing and centering constants. In this case we also assume that $F$ belongs to the maximum domain of attraction (MDA) of the Gumbel or Fréchet distributions, and we borrow the known normalizing and centering constants from these MDA. Thus, in the case of the MDA of the Gumbel distribution the maxima of $(S_{ni})$ may converge to the Gumbel distribution due to two distinct mechanisms: the normal approximation at medium-high thresholds or the subexponential approximation at high-level thresholds. In the case of the MDA of the Fréchet distribution two distinct approximations are possible: Gumbel approximation at medium-high thresholds and Fréchet approximation at high-level thresholds provided the distribution has finite second moment. If this condition is not satisfied only the Fréchet approximation is possible.

The paper is organized as follows. In Section 2 we introduce the necessary notions for this paper: subexponential and regularly varying distributions (Section 2.1), maximum domain of attraction and relevant distributions in it (Section 2.2), point process convergence of triangular arrays toward Poisson random measures (Section 2.3), precise large deviations (Section 2.4). Due to the importance of the latter topic we devote Section 3 to it and collect some of the known precise large deviation results in the case when the moment generating function is finite in some neighborhood of the origin and for subexponential distributions. The main results of this paper are formulated in Section 4. Based on the large deviation results of Section 3 we give sufficient conditions for the point process convergence relation (1.1) to hold and we clarify which rates of growth are possible for $p_{n}\to\infty$. In particular, we consider the case when $p_{n}$ in (1.1) is replaced by $k_{n}=[n/r_{n}]$ for some integer sequence $r_{n}\to\infty$ and $n$ is replaced by $r_{n}$. This means that we are interested in (1.1) when $S_{ni}=S_{r_{n}i}-S_{r_{n}(i-1)}$, $i=1,\ldots,k_{n}$, are iid block sums. We also discuss extensions of these results to stationary regularly varying sequences (Section 4.3.3) and iid multivariate regularly varying sequences (Section 4.3.4).

2. Preliminaries I

2.1. Subexponential and regularly varying distributions

We are interested in the class $\mathcal{S}$ of subexponential distributions $F$, i.e., it is a distribution supported on $[0,\infty)$ such that for any $n\geq 2$,

[TABLE]

For an encyclopedic treatment of subexponential distributions, see Foss et al. [10]. In insurance mathematics, ${\mathcal{S}}$ is considered a natural class of heavy-tailed distributions. In particular, $F$ does not have a finite moment generating function; see Embrechts et al. [8], Lemma 1.3.5.

The regularly varying distributions are another class of heavy-tailed distributions supported on ${\mathbb{R}}$. We say that $X$ and its distribution $F$ are regularly varying with index $\alpha>0$ if there are a slowly varying function $L$ and constants $p_{\pm}$ such that $p_{+}+p_{-}=1$ and

[TABLE]

A non-negative regularly varying $X$ is subexponential; see [8], Corollary 1.3.2.

2.2. Maximum domains of attraction

We call a non-degenerate distribution $H$ an extreme value distribution if there exist constants $c_{n}>0$ and $d_{n}\in{\mathbb{R}}$, $n\geq 1$, such that the maxima $M_{n}=\max(X_{1},\ldots,X_{n})$ satisfy the limit relation

[TABLE]

In the context of this paper we will deal with two standard extreme value distributions: the Fréchet distribution $\Phi_{\alpha}(x)=\exp(-x^{-\alpha})$, $x>0$, and the Gumbel distribution $\Lambda(x)=\exp(-\exp(-x))$, $x\in{\mathbb{R}}$. As a matter of fact, the third type of extreme value distribution – the Weibull distribution – cannot appear since (2.2) is only possible for $X$ with finite right endpoint but a random walk is not bounded from above by a constant. We say that the distribution $F$ of $X$ is in the maximum domain of attraction of the extreme value distribution $H$ $(F\in{\rm MDA}(H))$.

Example 2.1**.**

A distribution $F\in{\rm MDA}(\Phi_{\alpha})$ for some $\alpha>0$ if and only if

[TABLE]

see [8], Section 3.3.1. Then

[TABLE]

where $(c_{n})$ can be chosen such that $n\,{\mathbb{P}}(X>c_{n})\to 1$.

Example 2.2**.**

A distribution $F$ with infinite right endpoint obeys $F\in{\rm MDA}(\Lambda)$ if and only if there exists a positive function $a(x)$ with derivative $a^{\prime}(x)\to 0$ as $x\to\infty$ such that

[TABLE]

see [8], Section 3.3.3. Then

[TABLE]

where $(d_{n})$ can be chosen such that $n\,{\mathbb{P}}(X>d_{n})\to 1$ and $c_{n}=a(d_{n})$.

The standard normal distribution $\Phi\in{\rm MDA}(\Lambda)$ and satisfies

[TABLE]

where $c_{n}=1/d_{n}$ and

[TABLE]

Since $d_{n}\sim\sqrt{2\log n}$ we can replace $c_{n}$ in (2.3) by $1/\sqrt{2\log n}$ while $d_{n}$ cannot be replaced by $\sqrt{2\log n}$.

The standard lognormal distribution (i.e., $X=\exp(Y)$ for a standard normal random variable $Y$) is also in ${\rm MDA}(\Lambda)$. In particular, one can choose

[TABLE]

see [8], p. 156.

The standard Weibull distribution has tail $\overline{F}(x)=\exp(-x^{-\tau})$, $x>0$, $\tau>0$. We consider a distribution $F$ on $(0,\infty)$ with a Weibull-type tail $\overline{F}(x)\sim c\,x^{\beta}\exp(-\lambda x^{\tau})$ for constants $c,\beta,\lambda,\tau>0$. Then $F\in{\rm MDA}(\Lambda)$ and one can choose

[TABLE]

where $s_{n}=\lambda^{-1}\log n$; see [8], p. 155.

2.3. Point process convergence of independent triangular arrays

For further use we will need the following point process limit result (Resnick [29], Theorem 5.3).

Proposition 2.3**.**

Let $(X_{ni})_{n=1,2,\ldots;i=1,2,\ldots}$ be a triangular array of row-wise iid random elements on some state space $E\subset{\mathbb{R}}^{d}$ equipped with the Borel $\sigma$-field $\mathcal{E}$. Let $\mu$ be a Radon measure on $\mathcal{E}$. Then

[TABLE]

holds for some ${\rm PRM}(\mu)$ $N$ if and only if

[TABLE]

where $\stackrel{{\scriptstyle v}}{{\rightarrow}}$ denotes vague convergence on $E$.

2.4. Large deviations

Our main goal is to prove the point process convergence (1.1) for iid sequences $(S_{ni})$ of partial sum processes (${\mathbb{R}}$- or ${\mathbb{R}}^{d}$-valued), properly normalized and centered. It follows from Proposition 2.3 that this means to prove relations of the type

[TABLE]

provided $\mu(a,b]+\mu(a,\infty)<\infty$. Since $p=p_{n}\to\infty$ this means that {\mathbb{P}}\big{(}c_{n}^{-1}(S_{n}-d_{n})>a)\to 0 as $n\to\infty$. We will refer to these vanishing probabilities as large deviation probabilities. In Section 3 we consider some of the well-known precise large deviation results in heavy- and light-tail situations.

3. Preliminaries II: precise large deviations

In this section we collect some precise large deviation results in the light- and heavy-tailed cases.

3.1. Large deviations with normal approximation

We assume ${\mathbb{E}}[X]=0$, ${\rm var}(X)=1$ and write $\Phi$ for the standard normal distribution. We start with a classical result when $X$ has finite exponential moments.

Theorem 3.1** (Petrov’s theorem [26], Theorem 1 in Chapter VIII).**

Assume that the moment generating function ${\mathbb{E}}[\exp(h\,X)]$ is finite in some neighborhood of the origin. Then the following tail bound holds for $0\leq x=o(\sqrt{n})$:

[TABLE]

where $\lambda(t)=\sum_{k=0}^{\infty}a_{k}t^{k}$ is the Cramér series whose coefficients $a_{k}$ depend on the cumulants of $X$, and $\lambda(t)$ converges for sufficiently small values $|t|$.

Under the conditions of Theorem 3.1, uniformly for $x=o(n^{1/6})$,

[TABLE]

Theorem 7 in Chapter VIII of Petrov [26] considers the situation of Theorem 3.1 under the additional assumption that the cumulants of order $k=3,\ldots,r+2$ of $X$ vanish for some positive integer $r$. Then the coefficients $a_{0},\ldots,a_{r-1}$ in the series $\lambda(t)$ vanish, and it is not difficult to see that (3.1) holds uniformly for 0\leq x=o\big{(}n^{(r+1)/(2(r+3))}\big{)}.

In [26], Section VIII.3, one also finds necessary and sufficient conditions for (3.1) to hold in certain intervals. The following result was proved by S.V. Nagaev [21] for $x\in(0,\sqrt{(s/2-1)\log n})$ and improved by R. Michel [18] for $x\in(0,\sqrt{(s-2)\log n})$. The statement of the proposition is sharp under the given moment condition; see Theorem 3.5 below.

Proposition 3.2**.**

Assume that ${\mathbb{E}}[|X|^{s}]<\infty$ for some $s>2$. Then (3.1) holds uniformly for $0\leq x\leq\sqrt{(s-2)\log n}$.

3.2. Large deviations with normal/subexponential approximations

Cline and Hsing [4] (in an unpublished article) discovered that the subexponential class ${\mathcal{S}}$ of distributions exhibits a completely different kind of large deviation behavior:

Proposition 3.3** (Cline and Hsing [4]).**

We consider a distribution $F$ on $(0,\infty)$ with infinite right endpoint. Then the following statements hold.

$F\in{\mathcal{S}}$* if and only if*

[TABLE]

and there exists a sequence $\gamma_{n}\to\infty$ such that

[TABLE] 2. 2.

If $F\in{\mathcal{S}}$ then there exists a sequence $\gamma_{n}\to\infty$ such that

[TABLE]

Remark 3.4**.**

If $F$ satisfies (3.2) we say that $F$ is long-tailed, we write $F\in{\mathcal{L}}$. It is well known that $F\in{\mathcal{S}}$ implies $F\in\mathcal{L}$; see Embrechts et al. [8], Lemma 1.3.5 on p. 41. The converse is not true.

Proposition 3.3 shows that the subexponential class is the one for which heavy-tail large deviations are reasonable to study. Given that we know that $F$ is long-tailed, $F$ is subexponential if and only if a uniform large deviation relation of the type (3.4) holds.

Subexponential and normal approximations to large deviation probabilities were studied in detail in various papers. Among them, large deviations for iid regularly varying random variables are perhaps studied best. S.V. Nagaev [25] formulated a seminal result about the large deviations of a random walk $(S_{n})$ in the case of regularly varying $X$ with finite variance. He dedicated this theorem to his brother A.V. Nagaev who had started this line of research in the 1960s; see for example [22, 23].

Theorem 3.5** (Nagaev’s theorem [22, 25]).**

Consider an iid sequence $(X_{i})$ of random variables with ${\mathbb{E}}[X]=0$, ${\rm var}(X)=1$ and ${\mathbb{E}}[|X|^{2+\delta}]<\infty$ for some $\delta>0$. Assume that $\overline{F}(x)=x^{-\alpha}\,L(x)$, $x>0$, for some $\alpha>2$ and a slowly varying function $L$. Then for $x\geq\sqrt{n}$ as $n\to\infty$,

[TABLE]

In particular, if $X$ satisfies (2.1) with constants $p_{\pm}$, then for any positive constant $c_{1}<\alpha-2$

[TABLE]

and for any constant $c_{2}>\alpha-2$,

[TABLE]

Remark 3.6**.**

If $X$ is regularly varying with index $\alpha$, ${\mathbb{E}}[|X|^{s}]$ is finite (infinite) for $s<\alpha$ $(s>\alpha)$. Therefore the normal approximation (3.5) is in agreement with Proposition 3.2.

In the infinite variance regularly varying case this result is complemented by an analogous statement. It can be found in Cline and Hsing [4], Denisov et al. [6].

Theorem 3.7**.**

Consider an iid sequence $(X_{i})$ of regularly varying random variables with index $\alpha\in(0,2]$ satisfying (2.1). Assume ${\mathbb{E}}[X]=0$ if this expectation is finite. Choose $(a_{n})$ such that

[TABLE]

and $(\gamma_{n})$ such that $\gamma_{n}/a_{n}\to\infty$ as $n\to\infty$. For $\alpha=2$, also assume for sufficiently small $\delta>0$,

[TABLE]

Choose $(d_{n})$ such that

[TABLE]

Then the following large deviation result holds:

[TABLE]

Remark 3.8**.**

The normalization $(a_{n})$ is chosen such that $a_{n}^{-1}(S_{n}-d_{n})\stackrel{{\scriptstyle d}}{{\rightarrow}}Y_{\alpha}$ for an $\alpha$-stable random variable $Y_{\alpha}$, $\alpha\in(0,2]$. Therefore $\gamma_{n}^{-1}(S_{n}-d_{n})\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}0$. In the case $\alpha<2$, in view of Karamata’s theorem (see Bingham et al. [3]), it is possible to choose $(a_{n})$ according as $n\,{\mathbb{P}}(|X|>a_{n})\to 1$. The case $\alpha=2$ is delicate: in this case ${\rm var}(X)$ can be finite or infinite. In the former case, $(a_{n})$ is proportional to $\sqrt{n}$, in the latter case $(a_{n}/\sqrt{n})$ is a slowly varying sequence; see Feller [9] or Ibragimov and Linnik [15], Section II.6.

Normal and subexponential approximations to large deviation probabilities also exist for subexponential distributions that have all moments finite. Early on, this was observed by A.V. Nagaev [22, 23, 24]. Rozovskii [31] did not use the name of subexponential distribution, but the conditions on the tails of the distributions he introduced are “close” to subexponentiality; he also allowed for distributions $F$ supported on the whole real line. In particular, A.V. Nagaev and Rozovskii discovered that, in general, the $x$-regions where the normal and subexponential approximations hold are separated from each other. To make this precise, we call two sequences $(\xi_{n})$ and $(\psi_{n})$ separating sequences for the normal and subexponential approximations to large deviation probabilities if for an iid sequence $(X_{i})$ with variance 1,

[TABLE]

A.V. Nagaev and Rozovskii gave conditions under which $(\psi_{n})$ and $(\xi_{n})$ cannot have the same asymptotic order; i.e., one necessarily has $\psi_{n}/\xi_{n}\to\infty$. In particular, in the $x$-region $(\xi_{n},\psi_{n})$ neither the normal nor the subexponential approximation holds; Rozovskii [31] also provided large deviation approximations for ${\mathbb{P}}(S_{n}>x)$ for these regions involving $\overline{\Phi}(x/\sqrt{n})$ and a truncated Cramér series. Explicit expressions for $(\psi_{n})$ and $(\xi_{n})$ are in general hard to get. We focus on two classes of subexponential distributions where the separating sequences are known.

• •

Lognormal-type tails, we write $F\in{\rm LN}(\gamma)$: for some constants $\beta,\xi\in{\mathbb{R}}$, $\gamma>1$ and $\lambda,c>0$,

[TABLE]

In the notation ${\rm LN}(\gamma)$ we suppress the dependence on $\beta,\xi,\lambda,c$.

• •

Weibull-type tails, we write $F\in{\rm WE}(\tau)$: for some $\beta\in{\mathbb{R}}$, $\tau\in(0,1)$, $\lambda,c>0$.

[TABLE]

In the notation ${\rm WE}(\tau)$ we suppress the dependence on $\beta,\lambda,c$.

The name “Weibull-type tail” is motivated by the fact that the Weibull distribution $F$ with shape parameter $\tau\in(0,1)$ belongs to ${\rm WE}(\tau)$. Indeed, in this case $\overline{F}(x)=\exp(-\lambda x^{\tau})$, $x>0$, for positive parameters $\lambda$. Similarly, the lognormal distribution $F$ belongs to ${\rm LN}(2)$. This is easily seen by an application of Mill’s ratio: for a standard normal random variable $Y$,

[TABLE]

These classes of distributions have rather distinct tail behavior. It follows from the theory in Embrechts et al. [8], Sections 1.3 and 1.4, that membership of $F$ in ${\rm RV}(\alpha)$, ${\rm LN}(\gamma)$ or ${\rm WE}(\tau)$ implies $F\in\mathcal{S}$. The case ${\rm WE}(\tau)$, $0<\tau<1$, was already considered by A.V. Nagaev [23, 24].

For the heaviest tails when $F\in{\rm LN}(\gamma)$, $1<\gamma<2$ one can still choose $\xi_{n}=\psi_{n}$. This means that one threshold sequence separates the normal and subexponential approximations to the right tail ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>x)$. Rozovskii [31] discovered that the classes ${\rm LN}(\gamma)$, $\gamma\geq 2$ , and ${\rm LN}(\gamma)$, $1<\gamma<2$ have rather distinct large deviation properties. In the case $\gamma\geq 2$ one cannot choose $(\xi_{n})$ and $(\psi_{n})$ the same. The class LN($\gamma$) with $1<\gamma<2$ satisfies the conditions of Theorem 3b in Rozovskii [31] which implies that

[TABLE]

uniformly for $x$, where \gamma_{n}=\big{(}\lambda 2^{-\gamma+1}\big{)}^{1/2}n^{1/2}(\log n)^{\gamma/2}. For $\gamma=2$ the conditions of Theorem 3a in [31] are satisfied: with $g(x)=\lambda(\log x)^{2}-({\beta}+2)\log x-{\xi}\log\left(\log x\right)-\log c$ and as $n\to\infty$,

[TABLE]

Direct calculation shows that ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>\gamma_{n})\sim\exp(\lambda)\,n\,\overline{F}(\gamma_{n})$ while, uniformly for $x>\gamma_{n}h_{n}$, $h_{n}\to\infty$, we have that ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>x)\sim n\overline{F}(x)$.

It is interesting to observe that all but one class of subexponential distributions considered in Table 1 have the property that $c\,n\in(\psi_{n},\infty)$ for any $c>0$. The exception is ${\rm WE}(\tau)$ for $\tau\in(0.5,1)$. This fact turns the investigation of the tail probabilities ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>c\,n)$ into a complicated technical problem. The exponential (${\rm WE}(1)$) and superexponential (${\rm WE}(\tau)$), $\tau>1$, classes do not contain subexponential distributions. The corresponding partial sums exhibit the light-tailed large deviation behavior of Petrov’s Theorem 3.1. As a historical remark, Linnik [17] and S.V. Nagaev [21] determined lower separating sequences $(\xi_{n})$ for the normal approximation to the tails ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>x)$ under the assumption that $\overline{F}$ is dominated by the tail of a regular subexponential distribution from the table.

Denisov et al. [6] and Cline and Hsing [4] considered a unified approach to subexponential large deviation approximations for general subexponential and related distributions. In particular, they identified separating sequences $(\psi_{n})$ for the subexponential approximation of the tails ${\mathbb{P}}(S_{n}-{\mathbb{E}}[S_{n}]>x)$ for general subexponential distributions. Denisov et al. [6] also considered local versions, i.e., approximations to the tails ${\mathbb{P}}(S_{n}\in[x,x+T])$ for $T>0$ as $x\to\infty$.

4. Main results

4.1. Gumbel convergence via

normal approximations to large deviation probabilities for small $x$.

We assume that ${\mathbb{E}}[X]=0$ and ${\rm var}(X)=1$ and the large deviation approximation to the standard normal distribution $\Phi$ holds: for some $\gamma_{n}\to\infty$,

[TABLE]

We recall that $\Phi\in{\rm MDA}(\Lambda)$ and (2.3) holds. An analogous relation holds for the maxima of iid random walks $S_{n1}/\sqrt{n},\ldots,S_{np}/\sqrt{n}$ as follows from the next result.

Theorem 4.1**.**

Assume that (4.1) is satisfied for some $\gamma_{n}\to\infty$. Then

[TABLE]

holds for any integer sequence $p_{n}\to\infty$ such that $p_{n}<\exp(\gamma_{n}^{2}/2)$ and $(d_{p})$ is defined in (2.4). Moreover, for the considered $(p_{n})$, (4.2) is equivalent to either of the following limit relations:

1. (1)

For $\Gamma_{i}=E_{1}+\cdots+E_{i}$ and an iid standard exponential sequence $(E_{i})$ the following point process convergence holds on the state space ${\mathbb{R}}$

[TABLE]

where $N$ is ${\rm PRM}(-\log\Lambda)$ on ${\mathbb{R}}$. 2. (2)

Gumbel convergence of the maximum random walk

[TABLE]

Proof.

In view of Proposition 2.3 it suffices for $N_{p}\stackrel{{\scriptstyle d}}{{\rightarrow}}N$ to show that

[TABLE]

But this follows from (4.1) and the definition of $(d_{p})$ if we assume that $d_{p}+x/d_{p}<\gamma_{n}$, i.e., $p_{n}<\exp(\gamma_{n}^{2}/2)$ such that $p_{n}\to\infty$.

If $N_{p}\stackrel{{\scriptstyle d}}{{\rightarrow}}N$ a continuous mapping argument implies that

[TABLE]

On the other hand, for $x\in{\mathbb{R}}$ as $n\to\infty$,

[TABLE]

if and only if (4.2) holds. ∎

Remark 4.2**.**

If one replaces in (4.3) the quantities $(S_{ni}/\sqrt{n})_{i=1,\ldots,p}$ by iid standard normal random variables then this limit relation remains valid. This means that, under (4.1), e.g. under the assumption of a finite moment generating function in some neighborhood of the origin (see Section 3), the central limit theorem makes the tails of $(S_{ni}/\sqrt{n})_{i=1,\ldots,p}$ almost indistinguishable from those of the standard normal distribution. This is in stark contrast to subexponential distributions where the characteristics of $\overline{F}(x)$ show up in the tail ${\mathbb{P}}(S_{ni}/\sqrt{n}>x)$ for large values of $x$.

4.1.1. The extreme values of iid random walks.

Write

[TABLE]

for the ordered values of $S_{n1},\ldots,S_{np}$ The following result is immediate from Theorem 4.1.

Corollary 4.3**.**

Assume that the conditions of Theorem 4.1 hold. Then

[TABLE]

Moreover, if (4.1) also holds for the sequence $(-X_{i})$, then we have

Moreover, if there is $\gamma_{n}\to\infty$ such that \sup_{0\leq x<\gamma_{n}}\big{|}{{\mathbb{P}}(\pm S_{n}/\sqrt{n}>x)}/{\overline{\Phi}(x)}-1\Big{|}\to 0 as $n\to\infty$, then we have

[TABLE]

Proof.

We observe that $d_{p}/\sqrt{2\log p}\to 1$. Then (4.3) and the continuous mapping theorem imply that (4.4) holds for any fixed $k\geq 1$.

We observe that

[TABLE]

Of course, $P_{1}(x,y)\to\Lambda(x)$. On the other hand,

[TABLE]

The last step follows from a Taylor expansion of the logarithm and Theorem 4.1. This proves (4.5). ∎

4.1.2. Examples.

In this section we verify the assumptions of Theorem 4.1 for various classes of distributions $F$. We always assume ${\mathbb{E}}[X]=0$ and ${\rm var}(X)=1$.

Example 4.4**.**

Assume the existence of the moment generating function of $X$ in some neighborhood of the origin. Petrov’s Theorem 3.1 ensures (4.3) for $p\leq\exp(o(n^{1/3}))$.

Example 4.5**.**

Assume ${\mathbb{E}}[|X|^{s}]<\infty$ for some $s>2$. Proposition 3.2 ensures that (4.3) for $p\leq n^{(s-2)/2}$.

Example 4.6**.**

Assume that $X$ is regularly varying with index $\alpha>2$. Then we can apply Nagaev’s Theorem 3.5 with $\gamma_{n}=\sqrt{c\,\log n}$ for any $c<\alpha-2$ and (4.3) holds for $p\leq n^{c/2}$. This is in agreement with Example 4.5.

Example 4.7**.**

Assume that $X$ has a distribution in ${\rm LN}(\gamma)$ for some $\gamma>1$. From Table 1, $\gamma_{n}=o((\log n)^{\gamma/2}$, and (4.3) holds for $p\leq\exp(o((\log n)^{\gamma}))$

Example 4.8**.**

Assume that $F\in{\rm WE}(\tau)$, $0<\tau<1$. Table 1 yields $\gamma_{n}=o(n^{\tau/(2(2-\tau))})$ for $\tau\leq 0.5$, hence $p\leq\exp(o(n^{\tau/(2-\tau)}))$, and for $\tau\in(0.5,1)$, $\gamma_{n}=o(n^{1/6})$ and $p\leq\exp(o(n^{1/3}))$.

We summarize these examples in Table 2.

4.1.3. The extremes of the blocks of a random walk.

We consider a random walk $S_{n}$ with iid step sizes $X_{i}$ with ${\mathbb{E}}[X]=0$ and ${\rm var}(X)=1$, and with distribution $F$, and any integer sequence $r_{n}\to\infty$ such that $k_{n}=[n/r_{n}]\to\infty$ as $n\to\infty$. Set $S_{ni}=S_{r_{n}i}-S_{r_{n}(i-1)}$, i.e., this is the sum of the $i$th block $X_{r_{n}(i-1)+1},\ldots,X_{r_{n}i}$. Then we are in the setting of Theorem 4.1 if we replace $p_{n}$ by $k_{n}$ and $n$ by $r_{n}$. We are interested in the following result for the point process of the block sums of $S_{n}$ with length $r_{n}$ (see (4.3))

[TABLE]

This means we are looking for $(r_{n})$ such that $n/r_{n}<\exp(\gamma_{r_{n}}^{2}/2)$. This amounts to the following conditions on $(r_{n})$ in Table 3:

This table shows convincingly that, the heavier the tails, the larger we have to choose the block length $r_{n}$. Otherwise, the normal approximation does not function sufficiently well simultaneously for the block sums $S_{r_{n}i}-S_{r_{n}(i-1)}$, $i=1,\ldots,k_{n}$. In particular, in the regularly varying case we always need that $r_{n}$ grows polynomially.

Notice that we have from (4.6) in particular

[TABLE]

The normalization $d_{k_{n}}/\sqrt{r_{n}}$ is asymptotic to $\sqrt{(2\log k_{n})/r_{n}}$.

4.2. Gumbel convergence via the subexponential approximation to large deviation probabilities for

very large $x$

In this section we will exploit the subexponential approximation to large deviation probabilities for subexponential distributions $F$, i.e.,

[TABLE]

and we will also assume that $F\in{\rm MDA}(\Lambda)$; see Example 2.2 for the corresponding MDA conditions and the definition of the centering constants $(d_{n})$ and the normalizing constants $(c_{n})$. Then, in particular, $X$ has all moments finite. In this case, the Gumbel approximation of the point process of the $(S_{ni})$ is also possible.

Theorem 4.9**.**

Assume that $F\in{\rm MDA}(\Lambda)\cap\mathcal{S}$, the subexponential approximation (4.7) holds and for sufficiently large $n$ and an integer sequence $p_{n}\to\infty$,

[TABLE]

where $(d_{np})$ and $(c_{np})$ are the subsequences of $(d_{n})$ and $(c_{n})$, respectively, evaluated at $np$. Then

[TABLE]

holds. Moreover, (4.9) is equivalent to either of the following limit relations:

1. (1)

Point process convergence to a Poisson process on the state space ${\mathbb{R}}$

[TABLE]

where $N\sim{\rm PRM}(-\log\Lambda)$; see Theorem 4.1. 2. (2)

Gumbel convergence of the maximum random walk

[TABLE]

Proof.

If $d_{np}+x\,c_{np}>\gamma_{n}$ for every $x<0$ it holds for $x\in{\mathbb{R}}$. Therefore (4.7) applies. Since $F\in{\rm MDA}(\Lambda)\cap{\mathcal{S}}$ and by definition of $(c_{n})$ and $(d_{n})$ we have

[TABLE]

proving (4.9). Proposition 2.3 yields the equivalence of (4.10) and (4.9). The equivalence of (4.10) and (4.11) follows from a standard argument. ∎

Remark 4.10**.**

Since $a(x)$ defined in Example 2.2 has density $a^{\prime}(x)\to 0$ as $x\to\infty$ we have $a(x)/x\to 0$. On the other hand, $c_{n}=a(d_{n})$ and $d_{n}\to\infty$ since $F\in\mathcal{S}$. Therefore for any $x>0$,

[TABLE]

Hence (4.8) holds if $d_{np}\geq(1+\delta)\gamma_{n}$ for any small $\delta>0$ and large $n$.

4.2.1. The extreme values of iid random walks.

Relation (4.10) and a continuous mapping argument imply the following analog of Corollary 4.3. We use the same notation as in Section 4.1.1. One can follow the lines of the proof of Corollary 4.3.

Corollary 4.11**.**

Assume the conditions of Theorem 4.9. Then the following relation holds for $k\geq 1$,

[TABLE]

*as $n\to\infty$. *

4.2.2. Examples.

Theorem 4.9 applies to $F\in{\rm LN}(\gamma)$, $\gamma>1$, and $F\in{\rm WE}(\tau)$, $0<\tau<1$; see the discussion in Section 3.2. However, the calculation of the constants $(c_{n})$ and $(d_{n})$ is rather complicated for these classes of subexponential distributions. For illustration of the theory we restrict ourselves to two parametric classes of distributions where these constants are known.

Example 4.12**.**

We assume that $X$ has a standard lognormal distribution. From (2.5), Table 1 and Remark 4.10 we conclude that we need to verify the condition \exp\big{(}\sqrt{2\log(np)}\big{)}\geq h_{n}\sqrt{n}\log n for a sequence $(h_{n})$ increasing to infinity arbitrarily slowly. Calculation shows that it suffices to choose $p_{n}\to\infty$ such that p>\exp\big{(}(\log n)^{2}\big{)}.

Example 4.13**.**

We assume that $X$ has a Weibull distribution with tail $\overline{F}(x)=\exp({-x^{\tau}})$ for some $\tau\in(0,1)$. From (2.6) we conclude that $d_{np}\sim(\log np)^{1/\tau}$. In view of Remark 4.10 and Table 1 it suffices to verify that $(\log np)^{1/\tau}\geq h_{n}\,n^{1/(2-2\tau)}$ for a sequence $h_{n}\to\infty$ arbitrarily slowly. It holds if p>n^{-1}\,\exp\big{(}\big{(}h_{n}n^{1/(2-2\tau)}\big{)}^{\tau}\big{)}.

4.2.3. The extremes of the blocks of a random walk.

We appeal to the notation in Section 4.1.3. We are in the setting of Theorem 4.9 if we replace $p_{n}$ by $k_{n}$ and $n$ by $r_{n}$. We are interested in the following result for the point process of the block sums of $S_{n}$ with length $r_{n}$ (see (4.10))

[TABLE]

We need to verify condition (4.8) which turns into $d_{n}+c_{n}\,x>\gamma_{r_{n}}$. In view of Remark 4.10 it suffices to prove that $d_{n}>h_{n}\gamma_{r_{n}}$ for a sequence $h_{n}\to\infty$ arbitrarily slowly; see Table 1 for some $\gamma_{n}$-values.

We start with a standard lognormal distribution; see (2.5) for the corresponding $(c_{n})$ and $(d_{n})$. In particular, we need to verify

[TABLE]

A sufficient condition is $\exp(2\sqrt{2\log n})>\widetilde{h}_{n}r_{n}$ for a sequence $\widetilde{h}_{n}\to\infty$ arbitrarily slowly. We observe that the left-hand expression is a slowly varying function.

Next we consider a standard Weibull distribution for $\tau\in(0,1)$. The constants $(c_{n})$ and $(d_{n})$ are given in (2.6). In particular, we need to verify

[TABLE]

This holds if $(\log n)^{2(1-\tau)/\tau}h_{n}^{-2(1-\tau)}>r_{n}$. Again, this is a strong restriction on the growth of $(r_{n})$ and is in contrast to the regularly varying case where polynomial growth of $(r_{n})$ is possible; see Section 4.3.2.

4.3. Fréchet convergence via the subexponential approximations to

large deviation probabilities for large $x$

In this section we assume that $X$ is regularly varying with index $\alpha>0$ in the sense of (2.1). Throughout we choose a normalizing sequence $(a_{n})$ such that $n\,{\mathbb{P}}(|X|>a_{n})\to 1$ as $n\to\infty$. The following result is an analog of Theorems 4.1 and 4.9.

Theorem 4.14**.**

Assume that $X$ is regularly varying with index $\alpha>0$ and ${\mathbb{E}}[X]=0$ if the expectation is finite. Choose a sequence $(d_{n})$ such that

[TABLE]

We assume that $p_{n}\to\infty$ is an integer sequence which satisfies the additional conditions

[TABLE]

Then the following limit relation

[TABLE]

holds. Moreover, (4.17) is equivalent to

[TABLE]

where $(\Gamma_{i})$ is defined in Theorem 4.1 and $(q_{i})$ is an iid sequence of Bernoulli variables with distribution ${\mathbb{P}}(q_{i}=\pm 1)=p_{\pm}$ independent of $(\Gamma_{i})$.

Proof.

We start by verifying (4.17). Assume $\alpha<2$. Then for any sequence $p_{n}\to\infty$, $a_{np}/a_{n}\to\infty$. Therefore Theorem 3.7 and the definition of $(a_{np})$ yield

[TABLE]

If $\alpha>2$ the same result holds in view of Theorem 3.5 since we assume condition (4.16). If $\alpha=2$ we can again apply Theorem 3.7 with $\gamma_{n}=a_{np}$ and use (4.16).

We notice that the limit point process $N$ is ${\rm PRM}(\mu_{\alpha})$ with intensity

[TABLE]

An appeal to Proposition 2.3 shows that (4.17) and (4.18) are equivalent.

∎

Remark 4.15**.**

Assume $\alpha>2$. Since $a_{np}=(np)^{1/\alpha}\ell(np)$ for a slowly varying function $\ell$ and $\ell(x)\geq x^{-\gamma/\alpha}$ for any small $\gamma>0$ and sufficiently large $x$, (4.16) holds if $p\geq n^{(\alpha/2)-1+\gamma^{\prime}}$ for any choice of $\gamma^{\prime}>0$. Assume $\alpha=2$ and ${\rm var}(X)<\infty$. Then $a_{np}\sim c\,\sqrt{np}$ and (4.16) is satisfied for any sequence $p_{n}\to\infty$ and $\delta<1$. If ${\rm var}(X)=\infty$, $a_{np}=(np)^{1/2}\ell(np)$ for a slowly varying function $\ell$ and ${\mathbb{E}}[X^{2}(|X|\leq x)]$ is an increasing slowly varying function. Using Karamata bounds for slowly varying functions, we conclude that (4.16) holds if $p/n^{\gamma}\to\infty$ for any small $\gamma>0$.

4.3.1. The extreme values of iid random walks.

For simplicity, we assume $d_{n}=0$. Write $N_{p}^{+}$ for the restriction of $N_{p}$ to the state space $(0,\infty)$ and $S^{+}_{n,(1)}$ for the maximum of $(S_{n1})_{+},\ldots,(S_{np})_{+}$. We also write $\xi=\min\{i\geq 1:q_{i}=1\}$ and assume that $\xi$ is independent of $(\Gamma_{i})$. Then (4.18) and the continuous mapping theorem imply that

[TABLE]

Moreover, we have joint convergence of minima and maxima.

Corollary 4.16**.**

Assume the conditions of Theorem 4.14 and $d_{n}=0$. Then

[TABLE]

Proof.

We have

[TABLE]

∎

4.3.2. The extremes of the blocks of a random walk.

We appeal to the notation of Section 4.1.3 and apply Theorem 4.14 in the case when $n$ is replaced by some integer-sequence $r_{n}\to\infty$ such that $k_{n}=[n/r_{n}]\to\infty$ and $p_{n}$ is replaced by $k_{n}$. We also assume for simplicity that $d_{n}=0$. Observing that $a_{np}$ turns into $a_{r_{n}k_{n}}\sim a_{n}$, (4.18) turns into

[TABLE]

For simplicity, we assume $\alpha\neq 2$. If $\alpha<2$ no further restrictions on $(r_{n})$ are required. If $\alpha>2$ we have the additional growth condition $a_{n}>\sqrt{(\alpha-2+\delta)r_{n}\,\log r_{n}}$ for sufficiently large $n$. Since $a_{n}=n^{1/\alpha}\ell(n)$ for some slowly varying function $\ell$ this amounts to showing that $n^{2/\alpha}\ell^{2}(n)/(\alpha-2+\delta)>r_{n}\log r_{n}$. Since any slowly varying function satisfies $\ell(n)\geq n^{-\varepsilon}$ for any $\varepsilon>0$ and $n\geq n_{0}(\varepsilon)$ we get the following sufficient condition on the growth of $(r_{n})$: for any sufficiently small $\varepsilon>0$, $n^{2/\alpha-\varepsilon}>r_{n}$. This condition ensures that $(r_{n})$ is significantly smaller than $n$, and the larger $\alpha$ the more stringent this condition becomes.

An appeal to (4.20) yields in particular

[TABLE]

4.3.3. Extension to a stationary regularly varying sequence.

In view of classical theory (e.g. Feller [9]) $X$ is regularly varying with index $\alpha\in(0,2)$ if and only if $a_{n}^{-1}(S_{n}-d_{n})\stackrel{{\scriptstyle d}}{{\rightarrow}}\xi_{\alpha}$ for an $\alpha$-stable random variable $\xi_{\alpha}$ where one can choose $(a_{n})$ such that $n\,{\mathbb{P}}(|X|>a_{n})\to 1$ and $(d_{n})$ as in (3.8). For the sake of argument we also assume $d_{n}=0$; this is a restriction only in the case $\alpha=1$.

If $(r_{n})$ is any integer sequence such that $r_{n}\to\infty$ and $k_{n}=[n/r_{n}]\to 0$ then

[TABLE]

Moreover, since $a_{n}/a_{r_{n}}\to\infty$, Theorem 3.7 yields

[TABLE]

Classical limit theory for triangular arrays of the row-wise iid random variables $(S_{r_{n}i}-S_{r_{n}\,(i-1)})_{i=1,\ldots,k_{n}}$ (e.g. Petrov [26], Theorem 8 in Chapter IV) yields that (4.21) holds if and only if

[TABLE]

where $\mu_{\alpha}$ is defined in (4.19). We notice that (4.23) is equivalent to (4.22).

An alternative way of proving limit theory for the sum process $(S_{n})$ with an $\alpha$-stable limit $\xi_{\alpha}$ would be to assume the relations (4.23) and (4.24). This would be rather indirect and complicated in the case of iid $(X_{i})$. However, this approach has some merits in the case when $(X_{i})$ is a strictly stationary sequence with a regularly varying dependence structure, i.e., its finite-dimensional distributions satisfy a multivariate regular variation condition (see Davis and Hsing [5] or Basrak and Segers [1]), and a weak dependence assumption of the type

[TABLE]

holds. Then $a_{n}^{-1}S_{n}\stackrel{{\scriptstyle d}}{{\rightarrow}}\xi_{\alpha}$ if and only if $a_{n}^{-1}\sum_{i=1}^{k_{n}}S_{ni}\stackrel{{\scriptstyle d}}{{\rightarrow}}\xi_{\alpha}$ where $(S_{ni})_{i=1,\ldots,k_{n}}$ is an iid sequence with the same distribution as $S_{r_{n}}$. Condition (4.25) is satisfied under mild conditions on $(X_{i})$, in particular under standard mixing conditions such as $\alpha$-mixing. Thus one has to prove the conditions (4.23) and (4.24). In the dependent case the limit measure $\mu_{\alpha}$ has to be modified: the following analog of (4.22) holds: there exists a positive number $\theta_{X}$ such that

[TABLE]

The quantity $\theta_{X}$ has an explicit structure in terms of the so-called tail chain of the regularly varying sequence $(X_{i})$. It has interpretation as a cluster index in the context of the partial sum operation acting on $(X_{i})$. For details we refer to Mikosch and Wintenberger [20] and the references therein.

4.3.4. Extension to the multivariate regularly varying case.

Consider a sequence $({\bf X}_{i})$ of iid ${\mathbb{R}}^{d}$-valued random vectors with generic element ${\bf X}$, and define

[TABLE]

We say that ${\bf X}$ is regularly varying with index $\alpha>0$ and a Radon measure $\mu$ on ${\mathbb{R}}^{d}_{\boldsymbol{0}}={\mathbb{R}}^{d}\backslash\{{\boldsymbol{0}}\}$, and we write $\mathbf{X}\in\text{RV}(\alpha,\mu)$, if the following vague convergence relation is satisfied on ${\mathbb{R}}^{d}_{\boldsymbol{0}}$:

[TABLE]

and $\mu$ has the homogeneity property $\mu(t\,\cdot)=t^{-\alpha}\mu(\cdot)$, $t>0$. We will also use the sequential version of regular variation: for a sequence $(a_{n})$ such that $n{\mathbb{P}}(|{\bf X}|>a_{n})\to 1$, (4.26) is equivalent to

[TABLE]

For more reading on multivariate regular variation, we refer to Resnick [28, 29].

Hult et al. [14] extended Nagaev’s Theorem 3.5 to the multivariate case:

Theorem 4.17** (A multivariate Nagaev-type large deviation result).**

Consider an iid ${\mathbb{R}}^{d}$-valued sequence $({\bf X}_{i})$ with generic element ${\bf X}$. Assume the following conditions.

1. (1)

$\mathbf{X}\in\text{\rm RV}(\alpha,\mu)$. 2. (2)

The sequence of positive numbers $(x_{n})$ satisfies

[TABLE]

and, in addition,

[TABLE]

Then

[TABLE]

Remark 4.18**.**

Condition (4.27) requires that $n\,{\mathbb{E}}[{\bf X}]/a_{np}\to\bf 0$ for $\alpha>1$. It is always satisfied if ${\mathbb{E}}[{\bf X}]=\bf 0$. Now assume that the latter condition is satisfied if the expectation of ${\bf X}$ is finite. If $\alpha\in(0,2)$ we can choose any $(p_{n})$ such that $p_{n}\to\infty$. If $\alpha\geq 2$ and $(np)^{1/\alpha}/n^{0.5+\gamma/\alpha}\to\infty$, equivalently, $p/n^{\alpha/2-1+\gamma}\to\infty$ holds for any small $\gamma>0$ then (4.30) is satisfied.

The following result extends Theorem 4.14 to the multivariate case.

Theorem 4.19**.**

Assume that ${\bf X}$ satisfies the conditions of Theorem 4.17. Consider an integer sequence $p=p_{n}\to\infty$ and, in addition for $\alpha\geq 2$, that $x_{n}=a_{np}$ satisfies (4.30). Then the following limit relation holds

[TABLE]

where $({\bf S}_{ni})$ are iid copies of ${\bf S}_{n}$ and $N$ is ${\rm PRM}(\mu)$ on ${\mathbb{R}}^{d}_{\bf 0}$.

Proof.

In view of Proposition 2.3 it suffices to show that

[TABLE]

Assume $\alpha<2$. Then for any sequence $p_{n}\to\infty$, $a_{np}/a_{n}\to\infty$. Therefore Theorem  4.17 and the definition of $(a_{np})$ imply that for any $\mu$-continuity set $A\subset{\mathbb{R}}^{d}_{\boldsymbol{0}}$,

[TABLE]

If $\alpha\geq 2$ the same result holds by virtue of Theorem 4.17 and the additional condition (4.30). ∎

Example 4.20**.**

Write

[TABLE]

For vectors ${\bf x},{\bf y}\in{\mathbb{R}}^{d}$ with non-negative components, we write ${\bf x}\leq{\bf y}$ for the componentwise ordering, $[{\bf 0},{\bf x}]=\{{\bf y}:{\bf 0}\leq{\bf y}\leq{\bf x}\}$ and $[{\bf 0},{\bf x}]^{c}={\mathbb{R}}_{+}^{d}\backslash[\bf 0,{\bf x}]$. We have by Theorem 4.19,

[TABLE]

for the continuity points of the function $-\log H({\bf x})=\mu([{\bf 0},{\bf x}]^{c})$. If $\mu({\mathbb{R}}_{+}^{d}\backslash\{\bf 0\})$ is not zero $H$ defines a distribution on ${\mathbb{R}}_{+}^{d}$ with the property $-\log H(t{\bf x})=t^{-\alpha}(-\log H({\bf x}))$, $t>0$. The non-degenerate components of $H$ are in the type of the Fréchet distribution; $H$ is referred to as a multivariate Fréchet distribution with exponent measure $\mu$.

4.3.5. An extension to iid random sums.

In this section we consider an alternative random sum process:

[TABLE]

where $(\nu(t))_{t\geq 0}$ is a process of integer-valued non-negative random variables independent of the iid sequence $(X_{i})$ with generic element $X$ and finite expectation. Throughout we assume that $\lambda(t)={\mathbb{E}}[\nu(t)]$, $t\geq 0$, is finite but $\lim_{t\to\infty}\lambda(t)=\infty$. We also define

[TABLE]

In addition, we assume some technical conditions on the process $\nu$:

1. N1

$\nu(t)/\lambda(t)\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}1$, $t\to\infty$. 2. N2

There exist $\epsilon,\delta>0$ such that

[TABLE]

These conditions are satisfied for a wide variety of processes $\nu$, including the homogeneous Poisson process on $(0,\infty)$. Klüppelberg and Mikosch [16] proved the following large deviation result for the random sums $S(t)$. ([16] allow for the more general condition of extended regular variation.)

Theorem 4.21**.**

Assume that $\nu$ satisfies N1,N2 and is independent of the iid non-negative sequence $(X_{i})$ which is regularly varying with index $\alpha>1$. Then for any $\gamma>0$,

[TABLE]

The same method of proof as in the previous sections in combination with the large deviation result of Theorem 4.21 yields the following statement. As usual, we assume that $(a(t))$ is a function such that $t\,{\mathbb{P}}(X>a(t))\to 1$ as $t\to\infty$.

Corollary 4.22**.**

Assume the condition of Theorem 4.21. Let $(p(t))$ be an integer-valued function such that that $p(t)\to\infty$ as $t\to\infty$ and a growth condition is satisfied for every fixed $\gamma>0$ and sufficiently large $t\geq t_{0}$:

[TABLE]

Then the following limit relation holds for iid copies $S_{i}$ of the random sum process $S$:

[TABLE]

where $(\Gamma_{i})$ is defined in Theorem 4.1.

Proof.

In view of Proposition 2.3 the result is proved if we can show that as $t\to\infty$,

[TABLE]

But this follows by an application of Theorem 4.21 in combination with (4.32) and the regular variation of $X$. ∎

Remark 4.23**.**

Since $a(\lambda(t)p(t))=(\lambda(t)p(t))^{1/\alpha}\ell(\lambda(t)p(t))$ for a slowly varying function $\ell$ and $\ell(x)\geq x^{-\epsilon/\alpha}$ for any small $\epsilon>0$ and sufficiently large $x$, (4.32) holds if $p(t)\geq(\lambda(t))^{\alpha-1+\epsilon^{\prime}}$ for any choice of $\epsilon^{\prime}>0$ .

4.4. An extension: the index of the point process is random

Let $(P_{n})_{n\geq 0}$ be a sequence of positive integer-valued random variables. We assume that there exists a sequence of positive numbers $(p_{n})$ such that $p_{n}\to\infty$ and

[TABLE]

This condition is satisfied for wide classes of integer-valued sequences $(P_{n})$, including the renewal counting processes and (inhomogeneous) Poisson processes when calculated at the positive integers. In particular, for renewal processes $p_{n}\sim c\,n$ provided the inter-arrival times have finite expectation.

We have the following analog of Proposition 2.3.

Proposition 4.24**.**

Let $(X_{ni})_{n=1,2,\ldots;i=1,2,\ldots}$ be a triangular array of iid random variables assuming values in some state space $E\subset{\mathbb{R}}^{d}$ equipped with the Borel $\sigma$-field $\mathcal{E}$. Let $\mu$ be a Radon measure on ${\mathcal{E}}$. If the relation

[TABLE]

holds on $E$ then

[TABLE]

where $N$ is ${\rm PRM}(\mu)$ on $E$.

Proof.

We prove the result by showing convergence of the Laplace functionals. The arguments of a Laplace functional are elements of

[TABLE]

For $f\in C_{K}^{+}$ we have by independence of the $(X_{ni})$,

[TABLE]

In view of (4.33) there is a real sequence $\epsilon_{n}\downarrow 0$ such that

[TABLE]

Then

[TABLE]

By (4.35) we have $I_{1}\leq{\mathbb{P}}(A_{n}^{c})\to 0$ as $n\to\infty$ while

[TABLE]

In view of Proposition 2.3 and (4.34)

[TABLE]

The right-hand side is the Laplace functional of a ${\rm PRM}(\mu)$. Now an application of dominated convergence to $I_{2}$ in (4.36) yields the desired convergence result. ∎

An immediate consequence of this result is that all point process convergences in Section 4 remain valid if the point processes $N_{p}$ are replaced by their corresponding analogs $\widetilde{N}_{p}$ with a random index sequence $(P_{n})$ independent of $(S_{ni})$ and satisfying (4.33). Moreover, the growth rates for $p_{n}\to\infty$ remain the same.

4.5. Extension to the tail empirical process

We assume that $(S_{ni})$ are iid copies of a real-valued random walk $(S_{n})$. Instead of the point processes considered in the previous sections one can also study the tail empirical process

[TABLE]

where $k=k_{n}\to\infty$, $p=p_{n}\to\infty$ and $p_{n}/k_{n}\to\infty$, and $(c_{n})$ and $(d_{n})$ are suitable normalizing and centering constants. To illustrate the theory we consider two examples.

Example 4.25**.**

Assume the conditions and notation of Theorem 4.1. In this case, choose $c_{n}=1/d_{n}$. Then

[TABLE]

provided $p/k<\exp(\gamma_{n}^{2}/2)$. It is not difficult to see that

[TABLE]

Similarly, assume the conditions and the notation of Theorem 4.14 and consider

[TABLE]

Then for $x>0$ as $n\to\infty$,

[TABLE]

provided the modified sequence $p_{n}/k_{n}\to\infty$ satisfies the conditions imposed on $(p_{n})$ in Theorem 4.14. We notice that the values of $\mu_{\alpha}$ on $(-\infty,-x]$ and $(x,\infty)$ determine a Radon measure on ${\mathbb{R}}\backslash\{0\}$. From these relations we conclude that $N_{p}\stackrel{{\scriptstyle{\mathbb{P}}}}{{\rightarrow}}\mu_{\alpha}$. Then, following the lines of Resnick and Stărică [30], Proposition 2.3, one can for example prove consistency of the Hill estimator based on the sample $(S_{ni})_{i=1,\ldots,p}$: assuming for simplicity $d_{n}=0$, $p_{+}>0$, we write $S_{n,(1)}\geq\cdots\geq S_{n,(k)}$ for the $k$ largest values. Then

[TABLE]

4.6. Some related results

The largest values of sequences of iid normalized and centered partial sum processes play a role in the context of random matrix theory which is also the main motivation for the present work. Consider a double array $(X_{it})$ of iid regularly varying random variables with index $\alpha\in(0,4)$ (see (2.1)) and generic element $X$, and also assume that ${\mathbb{E}}[X]=0$ if this expectation is finite. Consider the data matrix

[TABLE]

and the corresponding sample covariance matrix ${\bf X}{\bf X}^{\top}=(S_{ij})$. Heiny and Mikosch [12] proved that

[TABLE]

where $\|A\|_{2}$ denotes the spectral norm of a $p\times p$ symmetric matrix $A$, ${\rm diag}(A)$ consists of the diagonal of $A$, $(a_{k})$ is any sequence satisfying $k\,{\mathbb{P}}(|X|>a_{k})\to 1$ as $k\to\infty$, and $p_{n}=n^{\beta}\ell(n)$ for some $\beta\in(0,1]$ and a slowly varying function $\ell$. Write $\lambda_{(1)}(A)\geq\cdots\geq\lambda_{(p)}(A)$ for the ordered eigenvalues of $A$. According to Weyl’s inequality (see Bhatia [2]), the eigenvalues of ${\bf X}{\bf X}^{\top}$ satisfy the relation

[TABLE]

But of course, $\lambda_{(i)}({\rm diag}({\bf X}{\bf X}^{\top}))$ are the ordered values of the iid partial sums $S_{ii}=\sum_{t=1}^{n}X_{it}^{2}$, $i=1,\ldots,p$. In view of (4.38) the asymptotic theory for the largest eigenvalues of the normalized sample covariance matrix $a_{np}^{-2}{\bf X}{\bf X}^{\top}$ (which also needs centering for $\alpha\in(2,4)$) are determined through the Fréchet convergence of the processes with points $(a_{np}^{-2}S_{ii})_{i=1,\ldots,p}$. Moreover, (4.38) implies the Fréchet convergence of the point processes of the normalized and centered eigenvalues of the sample covariance matrix.

The large deviation approach also works for proving limit theory for the point process of the off-diagonal elements of ${\bf X}{\bf X}^{\top}$ provided $X$ has sufficiently high moments. Heiny et al. [13] prove Gumbel convergence for the point process of the off-diagonal elements $(S_{ij})_{1\leq i<j\leq p}$. The situation is more complicated because the points $S_{ij}$ are typically dependent. Multivariate extensions of the normal large deviation approximation $0.5p^{2}{\mathbb{P}}(d_{p^{2}/2}(S_{12}-d_{p^{2}/2})>x)\to\exp(-x)$ show that the point process of the standardized $(S_{ij})$ has the same limit Poisson process as if the $S_{ij}$ were independent. Moreover, [13] show that the point process of the diagonal elements $(S_{ii})$ (under suitable conditions on the rate of $p_{n}\to\infty$ and under ${\mathbb{E}}[|X|^{s}]<\infty$ for $s>4$) converges to ${\rm PRM}(-\log\Lambda)$. This result indicates that the off-diagonal and diagonal entries of ${\bf X}{\bf X}^{\top}$ exhibit very similar extremal behavior. This is in stark contrast to the aforementioned results in [12] where the diagonal entries have Fréchet extremal behavior.

Related results can also be found in Gantert and Höfelsauer [11] who consider real-valued branching random walks and prove a large deviation principle for the position of the right-most particle; see Theorem 3.2 in [11]. The position of the right-most particle is the maximum of a collection of a random number of dependent random walks. In this context, the authors also prove a related large deviation result under the assumption that the considered random walks are iid. They show that the maximum of these iid random walks stochastically dominates the maximum of the branching random walks; see Theorem 3.1 and Lemma 5.2 in [11]. An early comparison between maxima of branching and iid random walks was provided by Durrett [7].