2.1. Subexponential and regularly varying distributions

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

\begin{equation*}\mathbb{P}(S_n>x)\sim n\,\overline F(x),\quad {x\to\infty}.\end{equation*}

For an encyclopedic treatment of subexponential distributions, see Foss, Korshunov, and Zachary [Reference Foss, Korshunov and Zachary10]. 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, Klüppelberg, and Mikosch [Reference Embrechts, Klüppelberg and Mikosch8, 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

(2.1) \begin{equation}F({-}x)\sim p_-\,x^{-\alpha}\,L(x)\quad\mbox{and}\quad \overline F(x)\sim p_+\,x^{-\alpha}\,L(x),\quad {x\to\infty}.\end{equation}

A non-negative regularly varying X is subexponential (see [Reference Embrechts, Klüppelberg and Mikosch8, 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 $d_n\in\mathbb{R}$ , $n\ge 1$ , such that the maxima $M_n=\max(X_1,\ldots,X_n)$ satisfy the limit relation

(2.2) \begin{equation}c_n^{-1} (M_n-d_n) \ {\stackrel{d}{\rightarrow}}\ Y\sim H,\quad{n\to\infty}.\end{equation}

In the context of this paper we 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 (Reference Bhatia2.Reference Bhatia2) 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 ${(F\in\text{MDA}(H))}$ .

Example 2.1. A distribution $F\in\text{MDA}(\Phi_\alpha)$ for some $\alpha>0$ if and only if

\begin{equation*}\overline F(x)= \frac{L(x)}{x^\alpha},\quad x>0\end{equation*}

(see [Reference Embrechts, Klüppelberg and Mikosch8, Section 3.3.1]). Then

\begin{equation*}c_n^{-1} M_n \ {\stackrel{d}{\rightarrow}}\ Y\sim \Phi_\alpha,\quad {n\to\infty},\end{equation*}

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 $F\in\text{MDA}(\Lambda)$ if and only if there exists a positive function a(x) with derivative $a'(x)\to 0$ as ${x\to\infty}$ such that

\begin{equation*}\lim_{u\to\infty}\frac{\overline F(u+ a(u)\, x)}{\overline F(u)}= \text{e}^{-x},\quad x\in\mathbb{R}\end{equation*}

(see [Reference Embrechts, Klüppelberg and Mikosch8, Section 3.3.3]). Then

\begin{equation*}c_n^{-1} (M_n-d_n) \ {\stackrel{d}{\rightarrow}}\ Y\sim \Lambda,\quad {n\to\infty},\end{equation*}

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 $\Phi\in \text{MDA}(\Lambda)$ and satisfies

(2.3) \begin{equation}c_n^{-1} (M_n-d_n)\ {\stackrel{d}{\rightarrow}}\ Y\sim \Lambda,\quad{n\to\infty},\end{equation}

where $c_n=1/d_n$ and

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

Since $d_n\sim \sqrt{2\log n}$ , we can replace $c_n$ in (Reference Bhatia2.Reference Bingham, Goldie and Teugels3) 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 $\text{MDA}(\Lambda)$ . In particular, one can choose

(2.5) \begin{equation}c_n=d_n/\sqrt{2\log n}\quad\mbox{and}\quad d_n=\exp\bigg(\sqrt{2\log n} - \frac{\log\log n+\log 4\pi}{2(2\log n)^{1/2}}\bigg)\end{equation}

(see [Reference Embrechts, Klüppelberg and Mikosch8, page 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 $F\in \text{MDA}(\Lambda)$ and one can choose

(2.6) \begin{equation}c_n=({\lambda} \tau)^{-1} s_n^{1/\tau-1}\quad\mbox{and}\quad d_n=s_n^{1/\tau}+\frac 1\tau s_n^{1/\tau-1} \bigg(\frac{\beta}{{\lambda} \tau} \log s_n + \frac{\log c}{{\lambda}}\bigg),\end{equation}

where $s_n={\lambda}^{-1}\log n$ (see [Reference Embrechts, Klüppelberg and Mikosch8, page 155]).

2.3. Point process convergence of independent triangular arrays

For further use we will need the following point process limit result (Resnick [Reference Resnick29, Theorem 5.3]).

Proposition 2.1. Let ${(X_{ni})_{n=1,2,\ldots;i=1,2,\ldots}}$ be a triangular array of row-wise i.i.d. 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

\begin{equation*}\widetilde N_p= \sum_{i=1}^p\varepsilon_{X_{ni}} \ {\stackrel{d}{\rightarrow}}\ N,\quad {n\to\infty},\end{equation*}

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

\begin{equation*} p\,\mathbb{P}(X_{n1}\in\cdot) \ {\stackrel{v}{\rightarrow}}\ \mu(\!\cdot\!),\quad {n\to\infty},\end{equation*}

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

2.4. Large deviations

Our main goal is to prove the point process convergence (1.1) for i.i.d. sequences ${(S_{ni})}$ of partial sum processes ( $\mathbb{R}$ - or $\mathbb{R}^d$ -valued), properly normalized and centered. It follows from Proposition 2.1 that this means proving relations of the type

\begin{equation*}p\,\mathbb{P}(c_n^{-1}(S_{n}-d_n) \in (a,b])\to \mu(a,b]\quad\mbox{or}\quad p\,\mathbb{P}(c_n^{-1}(S_{n}-d_n) >a)\to \mu(a,\infty),\end{equation*}

provided $\mu(a,b]+\mu(a,\infty)<\infty$ . Since $p=p_n\to\infty$ , this means that $\mathbb{P}(c_n^{-1}(S_{n}-d_n) >a)\to 0$ 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-tailed situations.

3.1. Large deviations with normal approximation

We assume $\mathbb{E}[X]=0$ , $\text{var}(X)=1$ 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 [Reference Petrov26], Theorem 1, 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 $0\le x={\text{o}}(\sqrt{n})$ :

\begin{equation*}\frac{\mathbb{P}(S_n/\sqrt{n}>x)}{\overline \Phi(x)}= \exp\bigg(\frac{x^3}{\sqrt{n}}{\lambda}\bigg(\frac{x}{\sqrt{n}}\bigg)\bigg) \bigg[1+ O\bigg(\frac{x+1}{\sqrt{n}}\bigg)\bigg],\quad {n\to\infty}.\end{equation*}

where ${\lambda}(t)= \sum_{k=0}^\infty a_kt^k$ 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 $x={\text{o}}(n^{1/6})$ ,

(3.1) \begin{equation}\frac{\mathbb{P}(S_n/\sqrt{n}>x)}{\overline \Phi(x)}\to 1,\quad {n\to\infty}.\end{equation}

Theorem 7 in Chapter VIII of Petrov [Reference Petrov26] 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 $0\le x={\text{o}}(n^{(r+1)/(2(r+3))})$ .

In Section VIII.3 of [Reference Petrov26] we also find necessary and sufficient conditions for (3.1) to hold in certain intervals. The following result was proved by S. V. Nagaev [Reference Nagaev21] for $x\in (0,\sqrt{(s/2-1)\log n})$ and improved by Michel [Reference Michel18] for $x\in (0,\sqrt{(s-2)\log n})$ . The statement of the proposition is sharp under the given moment condition; see Theorem 3.2 below.

3.2. Large deviations with normal/subexponential approximations

Cline and Hsing [Reference Cline and Hsing4] (in an unpublished article) discovered that the subexponential class ${\mathcal S}$ of distributions exhibits a completely different kind of large deviation behavior.

Proposition 3.2 shows that the subexponential class is the one for which heavy-tailed 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 (3.3) holds.

Subexponential and normal approximations to large deviation probabilities have been studied in detail in various papers. Among them, large deviations for i.i.d. regularly varying random variables are perhaps studied best. S. V. Nagaev [Reference Nagaev25] 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 [Reference Nagaev22] and [Reference Nagaev23].

Theorem 3.2. (Nagaev’s theorem [Reference Nagaev22, Reference Nagaev25].) Consider an i.i.d. sequence ${(X_i)}$ of random variables with $\mathbb{E}[X]=0$ , $\text{var}(X)=1$ 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 $x\ge \sqrt{n}$ as ${n\to\infty}$ ,

\begin{equation*}\mathbb{P}(S_n>x)=\overline \Phi(x/\sqrt{n})\,(1+{\text{o}}(1))+ n\,\overline F(x)\,(1+{\text{o}}(1)).\end{equation*}

In particular, if X satisfies (2.1) with constants $p_\pm$ , then for any positive constant $c_1< \alpha-2$ ,

(3.4) \begin{equation}\sup_{1<x/\sqrt{n} < \sqrt{c_1\,\log n}}\bigg|\frac{\mathbb{P}(\!\pm S_n>x)}{\overline \Phi(x/\sqrt{n})}-1\bigg|\to 0,\quad {n\to\infty},\end{equation}

and for any constant $c_2>\alpha-2$ ,

\begin{equation*}\sup_{x/\sqrt{n}>\sqrt{c_2\,\log n}}\bigg|\frac{\mathbb{P}(\!\pm S_n>x)}{n\,\mathbb{P}(|X|>x)}-p_{\pm}\bigg|\to 0,\quad{n\to\infty}.\end{equation*}

In the infinite variance regularly varying case this result is complemented by an analogous statement. It can be found in Cline and Hsing [Reference Cline and Hsing4] and Denisov, Dieker, and Shneer [Reference Denisov, Dieker and Shneer6].

Theorem 3.3. Consider an i.i.d. 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

\begin{equation*}n\,\mathbb{P}(|X|>a_n) + \frac{n}{a_n^2}\mathbb{E}[X^2\,\textbf{1}(|X|\le a_n)]=1,\quad n=1,2,\ldots,\end{equation*}

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

\begin{equation*}\lim_{{n\to\infty}}\sup_{x>\gamma_n} \frac{n}{x^2}\,\frac{\mathbb{E}[X^2\,\textbf{1}(|X|\le x)]}{[n\,\mathbb{P}(|X|>x)]^\delta}=0.\end{equation*}

Choose ${(d_n)}$ such that

(3.5) \begin{equation} d_n= \begin{cases}0 & \alpha\in (0,1)\cup(1,2],\\n\;\mathbb{E} [X\,\textbf{1}(|X|\le a_n)] &\alpha=1.\end{cases}\end{equation}

Then the following large deviation result holds:

\begin{equation*}\sup_{x>\gamma_n}\bigg|\frac{\mathbb{P}(\!\pm(S_n-d_n)>x)}{n\,\mathbb{P}(|X|>x)}-p_\pm \bigg|\to 0,\quad{n\to\infty}.\end{equation*}

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 [Reference Nagaev22, Reference Nagaev23, Reference Nagaev24]. Rozovskii [Reference Rozovski31] 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 i.i.d. sequence ${(X_i)}$ with variance 1,

\begin{align*}&\sup_{x < \xi_n}\bigg|\frac{\mathbb{P}(S_n-\mathbb{E}[S_n]>x)}{\overline \Phi(x/\sqrt{n})}-1\bigg|\to 0,\\*&\sup_{x> \psi_n}\bigg|\frac{\mathbb{P}(S_n-\mathbb{E}[S_n]>x)}{n\,\mathbb{P}(X>x)}-1\bigg|\to 0,\quad {n\to\infty}.\end{align*}

A. V. Nagaev and Rozovskii gave conditions under which ${(\psi_n)}$ and ${(\xi_n)}$ cannot have the same asymptotic order; that is, we necessarily have $\psi_n/\xi_n\to \infty$ . In particular, in the x-region ${(\xi_n,\psi_n)}$ neither the normal nor the subexponential approximation holds; Rozovskii [Reference Rozovski31] 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.

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

   \begin{equation*}\overline F(x)\sim c\,x^{\beta}\,(\!\log x)^\xi\,\exp\!({-}\lambda\,(\!\log x)^\gamma),\quad x\to\infty.\end{equation*}
   In the notation $\text{LN}(\gamma)$ we suppress the dependence on $\beta,\xi,\lambda,c$ .

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

   \begin{equation*}\overline F(x)\sim c\,x^{\beta}\,\exp\!({-}\lambda\, x^{\tau}),\quad x\to\infty.\end{equation*}
   In the notation $\text{WE}(\tau)$ 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 $\text{WE}(\tau)$ . Indeed, in this case $\overline F(x)=\exp\!({-}\lambda x^\tau)$ , $x>0$ , for positive parameters ${\lambda}$ . Similarly, the lognormal distribution F belongs to $\text{LN}(2)$ . This is easily seen by an application of Mill’s ratio: for a standard normal random variable Y,

\begin{equation*}\overline F(x)= \mathbb{P}(Y>\log x)\sim \frac{\exp\!({-} (\!\log x)^2/2)}{\sqrt{2\pi}\log x},\quad {x\to\infty}.\end{equation*}

These classes of distributions have rather distinct tail behavior. It follows from the theory in Embrechts et al. [Reference Embrechts, Klüppelberg and Mikosch8, Sections 1.3 and 1.4] that membership of F in $\text{RV}(\alpha)$ , $\text{LN}(\gamma)$ or $\text{WE}(\tau)$ implies $F\in\mathcal S$ . The case $\text{WE}(\tau)$ , $0<\tau<1$ , was already considered by A. V. Nagaev [Reference Nagaev23, Reference Nagaev24].

For the heaviest tails when $F\in \text{LN}(\gamma)$ , $1<\gamma< 2$ , we 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 [Reference Rozovski31] discovered that the classes $\text{LN}(\gamma)$ , $\gamma \ge 2$ , and $\text{LN}(\gamma)$ , $1< \gamma< 2$ have rather distinct large deviation properties. In the case $\gamma\ge 2$ we 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 [Reference Rozovski31], which implies that

\begin{equation*} \mathbb{P}(S_n-\mathbb{E}[S_n]>x ) = \big[ \overline{\Phi}(x/\sqrt{n} )\textbf{1} ( x<\gamma_n ) + n \overline{F}(x) \textbf{1} ( x>\gamma_n ) \big](1+{\text{o}}(1))\end{equation*}

uniformly for x, where $\gamma_{n} = (\lambda2^{-\gamma+1} )^{1/2} n^{1/2}(\!\log n)^{\gamma/2}$ . For $\gamma=2$ the conditions of Theorem 3a in [Reference Rozovski31] are satisfied: with $g(x) = \lambda (\!\log x)^{2} - ({\beta}+2) \log x -{\xi} \log ( \log x ) - \log c $ and as ${n\to\infty}$ ,

\begin{equation*} \mathbb{P}(S_n-\mathbb{E}[S_n]>x) = \big[ \overline{\Phi}(x/\sqrt{n} )\,\textbf{1}(x<\gamma_n)+ n \overline{F}(x) \text{e}^{\gpfrac{n(g'(x))^{2}}{2}}\textbf{1}(x>\gamma_n)\big](1+{\text{o}}(1)) .\end{equation*}

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 $\mathbb{P}(S_n-\mathbb{E}[S_n]>x ) \sim n \overline{F}(x)$ .

Table 1:

Separating sequences ${(\xi_n)}$ and ${(\psi_n)}$ for the normal and subexponential approximations of $\mathbb{P}(S_n-\mathbb{E}[S_n]>x)$ . We also assume $\text{var}(X)=1$ . Here ${(h_n), (\widetilde h_n)}$ are any sequences converging to infinity. For completeness, we also include the regularly varying class $\text{RV}(\alpha)$ . The table is taken from Mikosch and Nagaev [Reference Mikosch and Nagaev19].

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 $\text{WE}(\tau)$ 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 ( $\text{WE}(1)$ ) and superexponential ( $\text{WE}(\tau)$ ), $\tau> 1$ , 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 [Reference Linnik17] and S. V. Nagaev [Reference Nagaev21] 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. [Reference Denisov, Dieker and Shneer6] and Cline and Hsing [Reference Cline and Hsing4] 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. [Reference Denisov, Dieker and Shneer6] also considered local versions, i.e. approximations to the tails $\mathbb{P}(S_n\in [x,x+T])$ for $T>0$ as ${x\to\infty}$ .

4.1. Gumbel convergence via normal approximations to large deviation probabilities for small x

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

(4.1) \begin{equation}\sup_{0\le x < \gamma_n}\bigg|\frac{\mathbb{P}(S_n/\sqrt{n}>x)}{\overline \Phi(x)}-1\bigg|\to 0,\quad {n\to\infty}.\end{equation}

We recall that $\Phi\in \text{MDA}(\Lambda)$ and (2.3) holds. An analogous relation holds for the maxima of i.i.d. 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

(4.2) \begin{equation}p\,\mathbb{P}\bigg( \frac{S_n}{\sqrt{n}} >d_p+x/d_p\bigg)\to \text{e}^{-x},\quad {n\to\infty},\quad x\in\mathbb{R},\end{equation}

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 i.i.d. standard exponential sequence ${(E_i)}$ , the following point process convergence holds on the state space $\mathbb{R}$ :

   (4.3) \begin{equation}N_p=\sum_{i=1}^p \varepsilon_{d_p\,(\gfrac{S_{ni}}{\sqrt{n}} -d_p)}\ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty \varepsilon_{-\log \Gamma_i},\end{equation}
   where N is $\text{PRM}({-}\log \Lambda)$ on $\mathbb{R}$ .

2. (2) Gumbel convergence of the maximum random walk:

   \begin{equation*}d_p\max_{i=1,\ldots,p} (S_{ni}/\sqrt{n}-d_p)\ {\stackrel{d}{\rightarrow}}\ Y\sim\Lambda,\quad {n\to\infty}.\end{equation*}

Proof. In view of Proposition 2.1, it suffices for $N_p \ {\stackrel{d}{\rightarrow}}\ N$ to show that

\begin{equation*}p\,\mathbb{P}\bigg( d_{p} \bigg(\frac{S_n}{\sqrt{n}}-d_p\bigg) >x\bigg) =p\,\mathbb{P}\bigg( \frac{S_n}{\sqrt{n}} >d_p+x/d_p\bigg) \sim p\,\overline \Phi(d_p+x/d_p)\to \text{e}^{-x},\quad x\in\mathbb{R}.\end{equation*}

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 $N_p \ {\stackrel{d}{\rightarrow}}\ N$ , a continuous mapping argument implies that

\begin{align*}\mathbb{P}(N_p(x,\infty)=0)&=\mathbb{P}\Big(\max_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}-d_p)\le x\Big)\\*&\to\mathbb{P}(N(x,\infty)=0)\\*&= \Lambda(x),\quad x\in\mathbb{R},\quad{n\to\infty}.\end{align*}

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

\begin{align*}\mathbb{P}\Big(\max_{i=1,\ldots,p} d_p(S_{ni}/\sqrt{n}-d_p)\le x\Big)&= \bigg(1- \frac {p\,\mathbb{P}(d_p (S_{n1}/\sqrt{n}-d_p)>x)}{p}\bigg)^p \\*&\to \exp\!({-}\text{e}^{-x}),\end{align*}

if and only if (4.2) holds.

4.1.1. The extreme values of i.i.d. random walks

Write

\begin{equation*}S_{n,(p)}\le \cdots \le S_{n,(1)}\end{equation*}

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

Corollary 4.1. Assume that the conditions of Theorem 4.1hold. Then

(4.4) \begin{equation}\sqrt{2\log p}\,\bigg(\frac{S_{n,(1)}}{\sqrt{n}} -d_p,\ldots ,\frac{S_{n,(k)}}{\sqrt{n}} -d_p \bigg) \ {\stackrel{d}{\rightarrow}}\ ({-} \log \Gamma_1,\ldots,-\log \Gamma_k) ,\quad{n\to\infty}.\end{equation}

Moreover, if there is $\gamma_{n} \to \infty $ such that

\begin{equation*} \sup_{0\le x < \gamma_n}|{\mathbb{P}( \pm S_n/\sqrt{n}>x)}/{\overline \Phi(x)}-1|\to 0\quad\text{as ${n\to\infty}$,} \end{equation*}

then we have

(4.5) \begin{align}&\mathbb{P}\Big(\max_{i=1,\ldots,p} d_p(S_{ni}/\sqrt{n}-d_p)\le x,\min_{i=1,\ldots,p} d_p(S_{ni}/\sqrt{n}+d_p)\le y\Big)\notag\\*&\quad \to\Lambda(x) (1-\Lambda({-}y)),\quad x,y\in\mathbb{R},\quad {n\to\infty}.\end{align}

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 $k\ge 1$ .

We observe that

\begin{align*}& \mathbb{P}\Big(\max_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}-d_p )\le x,\min_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}+d_p )\le y\Big) \\* &\quad =\mathbb{P}\Big(\max_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}-d_p )\le x\Big)\\*&\quad\quad\, -\mathbb{P}\Big(\max_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}-d_p )\le x,\min_{i=1,\ldots,p} d_p (S_{ni}/\sqrt{n}+d_p )>y\Big)\\*&\quad =P_1(x,y)-P_2(x,y).\end{align*}

Of course, $P_1(x,y)\to \Lambda(x)$ . On the other hand,

\begin{align*}P_2(x,y)&=\mathbb{P}\bigg(\bigcap_{i=1}^p \{ S_{ni}/\sqrt{n}\le d_p+x/d_p,S_{ni}/\sqrt{n}>-d_p+y/d_p \}\bigg)\\*&= ( \mathbb{P} (\!-d_p+y/d_p < S_{n1}/\sqrt{n} \le d_p+x/d_p ) )^{p}\\ &= \exp\!( p \, \log ( 1- \mathbb{P} ( S_{n1}/\sqrt{n} > d_p+x/d_p ) -\mathbb{P} ( S_{n1}/\sqrt{n} \le -d_p+y/d_p ) ) )\\ &\to \exp\!({-}(\text{e}^{-x}+\text{e}^{y}) )\\* &=\Lambda(x)\Lambda({-}y).\end{align*}

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 $\text{var}(X)=1$ .

Example 4.1. 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 $p\le \exp\!({\text{o}}(n^{1/3}))$ .

Example 4.2. Assume $\mathbb{E}[|X|^s]<\infty$ for some $s>2$ . Proposition 3.1 ensures that (4.3) for $p\le n^{(s-2)/2}$ .

Example 4.3. Assume that X is regularly varying with index $\alpha>2$ . Then we can apply Nagaev’s Theorem 3.2 with $\gamma_n= \sqrt{c\,\log n}$ for any $c<\alpha-2$ and (4.3) holds for $p\le n^{c/2}$ . This is in agreement with Example 4.2.

Example 4.4. Assume that X has a distribution in $\text{LN}(\gamma)$ for some $\gamma>1$ . From Table 1, $\gamma_n={\text{o}}((\!\log n )^{\gamma/2}$ , and (4.3) holds for $p\le \exp\!( o((\!\log n )^{\gamma}))$ .

Example 4.5. Assume that $F\in \text{WE}(\tau)$ , $0<\tau<1$ . Table 1 yields $\gamma_n={\text{o}}(n^{\tau/(2(2-\tau))})$ for $\tau\le 0.5$ , hence $p\le \exp\!({\text{o}}(n^{\tau/(2-\tau)}))$ , and for $\tau\in (0.5,1)$ , $\gamma_n={\text{o}}(n^{1/6})$ and $p\le \exp\!({\text{o}}(n^{1/3}))$ .

We summarize these examples in Table 2.

Table 2:

Upper bounds for p.

4.1.3. The extremes of the blocks of a random walk

We consider a random walk $S_n$ with i.i.d. step sizes $X_i$ with $\mathbb{E}[X]=0$ and $\text{var}(X)=1$ , 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 ith 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)):

(4.6) \begin{equation}N_{k_n}=\sum_{i=1}^{k_n} \varepsilon_{d_{k_n}\, (\gfrac{ (S_{r_n i} -S_{r_n(i-1)})}{\sqrt{r_n}} -d_{k_n} )} \ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty \varepsilon_{-\log \Gamma_i}.\end{equation}

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

Table 3:

Lower bounds on the block size $r_n$ .

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

\begin{equation*}\frac{d_{k_n}}{\sqrt{r_n}} \,\max_{i=1,\ldots,k_n} (S_{r_n i} -S_{r_n(i-1)}-\sqrt{r_n}\,d_{k_n} ) \ {\stackrel{d}{\rightarrow}}\ -\log \Gamma_1\sim \Lambda,\quad{n\to\infty}.\end{equation*}

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, that is,

(4.7) \begin{equation}\sup_{x>\gamma_n} \bigg| \frac{\mathbb{P}(S_n-\mathbb{E}[S_n]>x)}{n\,\mathbb{P}(X>x)}- 1\bigg|\to 0,\end{equation}

and we will also assume that $F\in\text{MDA}(\Lambda)$ ; 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 4.2. Assume that $F\in\text{MDA}(\Lambda)\cap \mathcal S$ , the subexponential approximation (4.7) holds, and for sufficiently large n and an integer sequence $p_n\to\infty$ ,

(4.8) \begin{equation}d_{np}+x\,c_{np}>\gamma_n , \quad for\ any\ \mbox{$x<0$,}\end{equation}

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

(4.9) \begin{equation}p\,\mathbb{P} (S_n-\mathbb{E}[S_n]>d_{np}+ x\, c_{np} ) \to \text{e}^{-x},\quad x\in\mathbb{R},\quad {n\to\infty}{.}\end{equation}

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 $\mathbb{R}$ :

   (4.10) \begin{equation}N_p= \sum_{i=1}^p\varepsilon_{c_{np}^{-1} (S_{ni}-\mathbb{E}[S_n]-d_{np})} \ {\stackrel{d}{\rightarrow}}\ N,\quad {n\to\infty},\end{equation}
   where $N\sim\text{PRM}({-}\log \Lambda)$ ; see Theorem 4.1.

2. (2) Gumbel convergence of the maximum random walk:

   (4.11) \begin{equation}\max_{i=1,\ldots,p} c_{np}^{-1} ((S_{ni}-\mathbb{E}[S_n])-d_{np} ) \ {\stackrel{d}{\rightarrow}}\ Y\sim \Lambda,\quad {n\to\infty}.\end{equation}

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

\begin{equation*}p\,\mathbb{P} (S_n-\mathbb{E}[S_n]>d_{np}+ x\, c_{np} ) \sim p\,n\,\mathbb{P}(X>d_{np} +x\,c_{np})\to \text{e}^{-x},\quad x\in\mathbb{R},\quad {n\to\infty},\end{equation*}

proving (4.9). Proposition 2.1 yields the equivalence of (4.10) and (4.9). The equivalence of (4.10) and (4.11) follows from a standard argument. □

Remark 4.2. Since a(x) defined in Example 2.2 has density $a'(x)\to 0$ 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 $x>0$ ,

\begin{equation*}d_{np}+x\,c_{np}= d_{np}\bigg(1+ x\,\frac{a(d_{np})}{d_{np}}\bigg)\sim d_{np}.\end{equation*}

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

4.2.1. The extreme values of i.i.d. random walks

Relation (4.10) and a continuous mapping argument imply the following analog of Corollary 4.1. We use the same notation as in Section 4.1.1. One can follow the lines of the proof of Corollary 4.1.

Corollary 4.2. Assume the conditions of Theorem 4.2. Then the following relation holds for $k\ge 1$ :

\begin{equation*}c_{np}^{-1} (S_{n,(1)}-\mathbb{E}[S_n] -d_{np},\ldots ,S_{n,(k)}-\mathbb{E}[S_n] -d_{np} ) \ {\stackrel{d}{\rightarrow}}\ ({-} \log \Gamma_1,\ldots,-\log \Gamma_k )\end{equation*}

as ${n\to\infty}$ .

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 4.2 if we replace $p_n$ with $k_n$ and n with $r_n$ . 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)):

\begin{equation*} N_{k_n}=\sum_{i=1}^{k_n} \varepsilon_{c_{n}^{-1}\, (S_{r_n i} -S_{r_n(i-1)} -\mathbb{E}[S_{r_n}]-d_{n} )} \ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty \varepsilon_{-\log \Gamma_i}.\end{equation*}

We need to verify condition (4.8) which turns into $d_n+c_n\,x>\gamma_{r_n}$ . In view of Remark 4.2 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

\begin{equation*}d_n=\exp\bigg(\sqrt{2\log n} - \frac{\log\log n+\log 4\pi}{2(2\log n)^{1/2}}\bigg)\ge h_n \sqrt{r_n} \log r_n.\end{equation*}

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

\begin{equation*}d_n\sim ( \log n)^{1/\tau}> h_n r_n^{1/(2-2\tau)}.\end{equation*}

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 4.2.

Theorem 4.3. 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

\begin{equation*} d_n= \begin{cases} 0 & \alpha\in (0,1)\cup(1,\infty),\\*n\;\mathbb{E} [X\,\textbf{1}(|X|\le a_n)] &\alpha=1.\end{cases}\end{equation*}

We assume that $p_n\to\infty$ is an integer sequence which satisfies the additional conditions

(4.12) \begin{equation} \begin{cases}a_{np}\ge \sqrt{(\alpha-2+\delta) n \log n}\mbox{ for some small $\delta>0$}& if\ \mbox{$\alpha>2$,}\\[7pt] \lim_{{n\to\infty}}\sup_{x>a_{np}} p^\delta \frac{n}{x^2}\,\mathbb{E}[X^2\,\textbf{1}(|X|\le x)]=0\mbox{ for some small $\delta>0$}&if\ \mbox{$\alpha=2$.}\end{cases}\end{equation}

Then we obtain the limit relation

(4.13) \begin{equation}p\, \mathbb{P}(\!\pm a_{np}^{-1} (S_n-d_n)>x)\to p_\pm x^{-\alpha},\quad x>0,\quad{n\to\infty}{.}\end{equation}

Moreover, (4.13) is equivalent to

(4.14) \begin{equation}N_p=\sum_{i=1}^p \varepsilon_{a_{np}^{-1}(S_{ni}-d_n)} \ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty\varepsilon_{q_i\,\Gamma_i^{-1/\alpha}},\end{equation}

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

Proof. We start by verifying (4.13). Assume $\alpha<2$ . Then, for any sequence $p_n\to\infty$ , $a_{np}/a_n\to \infty$ . Therefore Theorem 3.3 and the definition of ${(a_{np})}$ yield

\begin{equation*}p\, \mathbb{P}(\!\pm a_{np}^{-1} (S_n-d_n)>x)\sim p\,n\,\mathbb{P}(\!\pm X>a_{np}\,x)\sim p_\pm x^{-\alpha},\quad {n\to\infty}.\end{equation*}

If $\alpha>2$ , the same result holds in view of Theorem 3.2 since we assume condition (4.12). If $\alpha=2$ , we can again apply Theorem 3.3 with $\gamma_n=a_{np}$ and use (4.12).

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

(4.15) \begin{equation}\mu_\alpha({\text{d}} x)= |x|^{-\alpha-1} (p_+ \textbf{1}(x>0)+p_- \textbf{1}(x<0) )\,{\text{d}} x.\end{equation}

An appeal to Proposition 2.1 shows that (4.13) and (4.14) are equivalent. □

4.3.1. The extreme values of i.i.d. random walks

For simplicity, we assume $d_n=0$ . Let $N_p^+$ be the restriction of $N_p$ to the state space ${(0,\infty)}$ , and $S^+_{n,(1)}$ the maximum of ${(S_{n1})_+,\ldots,(S_{np})_+}$ . We also write $\xi=\min\{i\ge 1 \colon q_i=1\}$ and assume that $\xi$ is independent of ${(\Gamma_i)}$ . Then (4.14) and the continuous mapping theorem imply that

(4.16) \begin{equation}\mathbb{P} (N_p^+(x,\infty)=0 )=\mathbb{P} (a_{np}^{-1}S^+_{n,(1)}\le x )\ {\stackrel{d}{\rightarrow}}\ \mathbb{P} (\Gamma_\xi^{-1/\alpha}\le x )=\Phi_\alpha^{p_+}(x).\end{equation}

Moreover, we have joint convergence of minima and maxima.

Corollary 4.3. Assume the conditions of Theorem 4.3 and $d_n=0$ . Then

\begin{equation*}\lim_{{n\to\infty}}\mathbb{P}\Big(0<a_{np}^{-1}\max_{i=1,\ldots,p} S_{ni}\le x,-y<a_{np}^{-1}\min_{i=1,\ldots,p} S_{ni} \Big)= \Phi_\alpha^{p_+}(x)\Phi_\alpha^{p_-}(y),\quad x,y>0.\end{equation*}

Proof. We have

\begin{align*}\mathbb{P}\Big(a_{np}^{-1}\max_{i=1,\ldots,p} S_{ni}\le x,-y<a_{np}^{-1}\min_{i=1,\ldots,p} S_{ni}\Big)&=\mathbb{P} ( N_p ( (x,\infty)\cup ({-}\infty,-y] )=0 )\\*&\to \mathbb{P} ( N ( (x,\infty)\cup ({-}\infty,-y] )=0 )\\&= \exp ({-}( p_+x^{-\alpha}+ p_- y^{-\alpha}) )\\*&= \Phi_\alpha^{p_+}(x)\Phi_\alpha^{p_-}(y),\quad{n\to\infty}. \square\end{align*}

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 4.3 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 $a_{r_n k_n}\sim a_n$ , (4.14) turns into

\begin{equation*}N_{k_n}=\sum_{i=1}^{k_n} \varepsilon_{a_{n}^{-1}(S_{r_n\,i}-S_{r_n(i-1)})} \ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty\varepsilon_{q_i\,\Gamma_i^{-1/\alpha}}, \quad {n\to\infty}.\end{equation*}

For simplicity, we assume $\alpha\ne 2$ . 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 $\ell(n)\ge n^{-\varepsilon}$ for any $\varepsilon>0$ and $n\ge n_0(\varepsilon)$ , 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 (4.16) yields in particular

\begin{align*}\mathbb{P}\Big(a_{n}^{-1}\max_{i=1,\ldots,k_n} (S_{r_n\,i}- S_{r_n\,(i-1)})_+\le x \Big)\ {\stackrel{d}{\rightarrow}}\ \mathbb{P} (\Gamma_\xi^{-1/\alpha}\le x )&=\Phi_\alpha^{p_+}(x),\\*\mathbb{P}\Big(a_{n}^{-1}\max_{i=1,\ldots,k_n} |S_{r_n\,i}- S_{r_n\,(i-1)}|\le x \Big)\ {\stackrel{d}{\rightarrow}}\ \mathbb{P} (\Gamma_1^{-1/\alpha}\le x )&=\Phi_\alpha(x),\quad {n\to\infty}.\end{align*}

4.3.3. Extension to a stationary regularly varying sequence

In view of classical theory (e.g. Feller [Reference Feller9]), X is regularly varying with index $\alpha \in (0,2)$ if and only if $a_n^{-1}(S_n-d_n) \ {\stackrel{d}{\rightarrow}}\ \xi_\alpha$ 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 (3.5). 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

(4.17) \begin{equation}a_n^{-1}S_n= a_n^{-1}\sum_{i=1}^{k_n} (S_{r_n\,i}- S_{r_n\,(i-1)})+o_\mathbb{P}(1) \ {\stackrel{d}{\rightarrow}}\ \xi_{\alpha}.\end{equation}

Moreover, since $a_n/a_{r_n}\to\infty$ , Theorem 3.3 yields

(4.18) \begin{equation}\frac{\mathbb{P}(\!\pm a_n^{-1}S_{r_n}>x)}{r_n\,\mathbb{P}(|X|>a_n)}\sim \frac{\mathbb{P}(\!\pm X>x\,a_n)}{\mathbb{P}(|X|>a_n)}\to p_{\pm}\,x^{-\alpha},\quad x>0.\end{equation}

Classical limit theory for triangular arrays of the row-wise i.i.d. random variables ${(S_{r_ni}-S_{r_n\,(i-1)})_{i=1,\ldots,k_n}}$ (e.g. Petrov [Reference Petrov26, Theorem Reference Embrechts, Klüppelberg and Mikosch8, Chapter IV]) yields that (4.17) holds if and only if

(4.19) \begin{gather}k_n\,\mathbb{P}(a_n^{-1} S_{r_n}\in\cdot) \ {\stackrel{v}{\rightarrow}}\ \mu_\alpha(\!\cdot\!),\end{gather}

(4.20) \begin{align}\lim_{\delta\downarrow 0}\limsup_{{n\to\infty}}k_n \text{var} (a_n^{-1} S_{r_n}\textbf{1} (|S_{r_n}|\le \delta a_n ) )=0, \end{align}

where $\mu_\alpha$ is defined in (4.15). We notice that (4.19) is equivalent to (4.18).

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 (4.19) and (4.20). This would be rather indirect and complicated in the case of i.i.d. ${(X_i)}$ . However, this approach has some merits in the case when ${(X_i)}$ is a strictly stationary sequence with a regularly varying dependence structure, that is, its finite-dimensional distributions satisfy a multivariate regular variation condition (see Davis and Hsing [Reference Davis and Hsing5] or Basrak and Segers [Reference Basrak and Segers1]), and we have a weak dependence assumption of the type

(4.21) \begin{equation}\mathbb{E} [\!\exp (a_n^{-1} {\text{i}} t S_n ) ]- (\mathbb{E} [\!\exp (a_n^{-1} {\text{i}} t S_{r_n} ) ] )^{k_n}\to 0,\quad t\in\mathbb{R},\quad {n\to\infty}{.}\end{equation}

Then $a_n^{-1}S_n \ {\stackrel{d}{\rightarrow}}\ \xi_\alpha$ if and only if $a_n^{-1} \sum_{i=1}^{k_n} S_{ni}\ {\stackrel{d}{\rightarrow}}\ \xi_\alpha$ , where ${(S_{ni})_{i=1,\ldots,k_n}}$ is an i.i.d. sequence with the same distribution as $S_{r_n}$ . Condition (4.21) is satisfied under mild conditions on ${(X_i)}$ , in particular under standard mixing conditions such as $\alpha$ -mixing. Thus we have to prove conditions (4.19) and (4.20). In the dependent case the limit measure $\mu_\alpha$ has to be modified. The following analog of (4.18) holds: there exists a positive number $\theta_X$ such that

\begin{equation*}\frac{\mathbb{P}(\!\pm a_n^{-1}S_{r_n}>x)}{r_n\,\mathbb{P}(|X|>a_n)}\sim \theta_X\,\frac{\mathbb{P}(\!\pm X>x\,a_n)}{\mathbb{P}(|X|>a_n)}\to \theta_X\,p_{\pm}\,x^{-\alpha},\quad x>0.\end{equation*}

The quantity $\theta_X$ has an explicit structure in terms of the so-called tail chain of the regularly varying sequence ${(X_i)}$ . It can be interpreted as a cluster index in the context of the partial sum operation acting on ${(X_i)}$ . For details we refer to Mikosch and Wintenberger [Reference Mikosch and Wintenberger20] and the references therein.

4.3.4. Extension to the multivariate regularly varying case

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

\begin{equation*}\textbf{S}_0=\boldsymbol{0}, \quad \textbf{S}_n= \textbf{X}_1 + \cdots + \textbf{X}_n , \quad n\ge 1.\end{equation*}

We say that 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 $\textbf{X} \in \text{RV} (\alpha, \mu )$ if the following vague convergence relation is satisfied on $\mathbb{R}^{d}_{\boldsymbol{0}}$ :

(4.22) \begin{equation}\frac{\mathbb{P}(x^{-1} \textbf{X} \in\cdot)}{\mathbb{P}(| \textbf{X}|>x )} \ {\stackrel{v}{\rightarrow}}\ \mu(\!\cdot\!), \quad {x\to\infty} ,\end{equation}

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 $n\mathbb{P}(| \textbf{X}|>a_n)\to 1$ , (4.22) is equivalent to

\begin{equation*}n\,\mathbb{P}(a_{n}^{-1} \textbf{X} \in\cdot) \ {\stackrel{v}{\rightarrow}}\ \mu(\!\cdot\!), \quad {n\to\infty} .\end{equation*}

For more reading on multivariate regular variation, we refer to Resnick [Reference Resnick28, Reference Resnick29].

Hult, Lindskog, Mikosch, and Samorodnitsky [Reference Hult, Lindskog, Mikosch and Samorodnitsky14] extended Nagaev’s Theorem 3.2 to the multivariate case.

Theorem 4.4. (A multivariate Nagaev-type large deviation result.) Consider an i.i.d. $\mathbb{R}^{d}$ -valued sequence ${(\textbf{X}_i)}$ with generic element X. Assume the following conditions.

1. (1) $\textbf{X} \in \text{RV} (\alpha, \mu )$ .

2. (2) The sequence of positive numbers ${(x_n)}$ satisfies

   (4.23) \begin{equation} x_{n}^{-1} \textbf{S}_n \ {\stackrel{\mathbb{P}}{\rightarrow}}\ \boldsymbol{0}\quad \text{as } {n\to\infty},\end{equation}
   and, in addition,
   (4.24) \begin{equation} \begin{cases} \frac{x_{n}^{2}}{n \mathbb{E}[ | \textbf{X}|^{2} \textbf{1}( | \textbf{X}| \leq x_{n} ) ] \log x_n } \to \infty & \alpha=2 \, \mbox{ and } \, \mathbb{E}[ | \textbf{X}|^{2} ]=\infty,\\[9pt]\frac{x_{n}^{2}}{n \log n} \to \infty & \alpha>2\,\mbox{ or }\, [\alpha=2\,\mbox{ and } \,\mathbb{E}[ | \textbf{X}|^{2} ]<\infty].\end{cases}\end{equation}
   Then
   \begin{equation*} \frac{\mathbb{P}( x_{n}^{-1} \textbf{S}_n \in\cdot)}{n \mathbb{P}( |\textbf{X}| >x_{n})} \ {\stackrel{v}{\rightarrow}}\ \mu (\!\cdot\!),\quad {n\to\infty}.\end{equation*}

Remark 4.4. Condition (4.23) requires that $n\,\mathbb{E}[\textbf{X}]/a_{np}\to\bf0$ for $\alpha>1$ . It is always satisfied if $\mathbb{E}[\textbf{X}]=\bf0$ . Now assume that the latter condition is satisfied if the expectation of X is finite. If $\alpha\in (0,2)$ , we can choose any ${(p_n)}$ such that $p_n\to\infty$ . If $\alpha\ge 2$ 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 (4.24) is satisfied.

The following result extends Theorem 4.3 to the multivariate case.

Theorem 4.5. Assume that X satisfies the conditions of Theorem 4.4. Consider an integer sequence $p=p_n\to\infty$ and, in addition for $\alpha\ge 2$ , that $x_n=a_{np}$ satisfies (4.24). Then the following limit relation holds:

\begin{equation*} N _p=\sum_{i=1}^p \varepsilon_{a_{np}^{-1}\textbf{S}_{ni}} \ {\stackrel{d}{\rightarrow}}\ N,\end{equation*}

where ${(\textbf{S}_{ni})}$ are i.i.d. copies of $\textbf{S}_n$ and N is $\text{PRM}(\mu)$ on $\mathbb{R}^d_{\bf0}$ .

Proof. In view of Proposition 2.1 it suffices to show that

\begin{equation*}p\, \mathbb{P}( a_{np}^{-1} \textbf{S}_n \in\cdot) \ {\stackrel{v}{\rightarrow}}\ \mu(\!\cdot\!).\end{equation*}

Assume $\alpha<2$ . Then, for any sequence $p_n\to\infty$ , $a_{np}/a_n\to \infty$ . Therefore Theorem 4.4 and the definition of ${(a_{np})}$ imply that, for any $\mu$ -continuity set $A\subset \mathbb{R}^{d}_{\boldsymbol{0}} $ ,

\begin{equation*}p\, \mathbb{P}( a_{np}^{-1} \textbf{S}_n \in A)\sim p\,n\,\mathbb{P}( |\textbf{X}|>a_{np}) \,\mu(A)\to \mu(A),\quad {n\to\infty}.\end{equation*}

If $\alpha\ge 2$ , the same result holds by virtue of Theorem 4.4 and the additional condition (4.24). □

Example 4.8. Write

\begin{align*}\textbf{S}_{ni}&= (S_{ni}^{(1)},\ldots,S_{ni}^{(d)})^\top,\\*\textbf{M}_n&= \Big(\max_{i=1,\ldots,p}S_{ni}^{(1)},\ldots,\max_{i=1,\ldots,p}S_{ni}^{(d)}\Big)^\top= (M_n^{(1)},\ldots,M_n^{(d)} )^\top.\end{align*}

For vectors $\textbf{x},\textbf{y}\in\mathbb{R}^d$ with non-negative components, we write $\textbf{x}\le \textbf{y}$ for the componentwise ordering, $[{\bf0},\textbf{x}]=\{\textbf{y} \colon {\bf0}\le \textbf{y}\le \textbf{x}\}$ and $[{\bf0},\textbf{x}]^c= \mathbb{R}_+^d\backslash [\bf0,\textbf{x}]$ . We have by Theorem 4.5,

\begin{align*}\mathbb{P} ({\bf0}\le a_{np}^{-1}\textbf{M}_n\le\textbf{x} )&= \mathbb{P} (N_p([{\bf0},\textbf{x}]^c)=0 )\\ &\to \mathbb{P} (N([{\bf0},\textbf{x}]^c)=0 )\displaybreak\\ &= \exp ({-}\mu([{\bf0},\textbf{x}]^c) )\\ &=\!:\,H(\textbf{x}),\quad {n\to\infty},\end{align*}

for the continuity points of the function $-\log H(\textbf{x})=\mu([{\bf0},\textbf{x}]^c)$ . If $\mu(\mathbb{R}_+^d\backslash \{\bf0\})$ is not zero, H defines a distribution on $\mathbb{R}_+^d$ with the property $-\log H(t\textbf{x})=t^{-\alpha}({-}\log H(\textbf{x}))$ , $t>0$ . 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 i.i.d. random sums

In this section we consider an alternative random sum process:

\begin{equation*}S(t)=\sum_{i=1}^{ \nu(t)}X_{i} ,\quad t\ge 0,\end{equation*}

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

\begin{equation*} m(t)=\mathbb{E}[S(t)]=\mathbb{E}[X]\,\lambda(t) .\end{equation*}

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

1. N1 $\nu(t)/\lambda(t) \ {\stackrel{\mathbb{P}}{\rightarrow}}\ 1$ , $t\to\infty$ .

2. N2 There exist $\epsilon, \delta>0$ such that

   \begin{equation*}\lim_{t\to\infty}\sum_{k>(1+\delta) \lambda(t) } \mathbb{P} (\nu(t)>k)\, (1+\epsilon)^{k} =0 .\end{equation*}

These conditions are satisfied for a wide variety of processes $\nu$ , including the homogeneous Poisson process on ${(0,\infty)}$ . Klüppelberg and Mikosch [Reference Klüppelberg and Mikosch16] proved the following large deviation result for the random sums S(t) (allowing for the more general condition of extended regular variation).

Theorem 4.6. Assume that $\nu$ satisfies $\textbf{N1}$ and $\textbf{N2}$ and is independent of the i.i.d. non-negative sequence ${(X_i)}$ which is regularly varying with index $\alpha>1$ . Then, for any $ \gamma>0$ ,

\begin{equation*}\sup_{x\ge \gamma\lambda(t)} \bigg|\frac{\mathbb{P}(S(t)-m(t) > x)}{\lambda(t) \mathbb{P}(X>x)}-1\bigg|,\quad t\to\infty .\end{equation*}

The same method of proof as in the previous sections in combination with the large deviation result of Theorem 4.6 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 4.4. Assume the condition of Theorem 4.6. 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 $t\ge t_0$ :

(4.25) \begin{equation}a({\lambda(t) p(t) })\ge \gamma\, \lambda(t).\end{equation}

Then the following limit relation holds for i.i.d. copies $S_i$ of the random sum process S:

\begin{equation*}N_{p(t)}=\sum_{i=1}^{p(t)} \varepsilon_{\gfrac{(S_{i}(t)-m(t))}{(a( \lambda(t) p(t)))}}\ {\stackrel{d}{\rightarrow}}\ N=\sum_{i=1}^\infty\varepsilon_{\Gamma_i^{-1/\alpha}},\quad t\to\infty,\end{equation*}

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

Proof. In view of Proposition 2.1 the result is proved if we can show that as $t\to\infty$ ,

\begin{align*}&p(t) \, \mathbb{P}( (a(\lambda(t) p(t)))^{-1} ( S(t)- m(t))>x )\sim \lambda(t)\,p(t)\,\mathbb{P}( X>a(\lambda(t) p(t) )\,x)\to x^{-\alpha},\\*& p(t) \, \mathbb{P}( (a(\lambda(t) p(t)))^{-1} ( S(t)- m(t))<-x )\to 0,\quad x>0.\end{align*}

But this follows by an application of Theorem 4.6 in combination with (4.25) and the regular variation of X. □

Remark 4.5. Since $a(\lambda(t) p(t))=(\lambda(t)p(t))^{1/\alpha}\ell(\lambda(t)p(t))$ for a slowly varying function $\ell$ and $\ell(x)\ge x^{-\epsilon/\alpha}$ for any small $\epsilon>0$ and sufficiently large x, (4.25) holds if $p(t)\ge (\lambda(t))^{\alpha-1+\epsilon'}$ for any choice of $\epsilon'>0$ .

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

Let ${(P_n)_{n\ge 0}}$ 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

(4.26) \begin{equation} \frac{P_n}{p_n} \ {\stackrel{\mathbb{P}}{\rightarrow}}\ 1 ,\quad {n\to\infty}.\end{equation}

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 2.1.

Proposition 4.1. Let $(X_{ni})_{n=1,2,\ldots;i=1,2,\ldots}$ be a triangular array of i.i.d. 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

(4.27) \begin{equation}p_n \,\mathbb{P} (X_{n1}\in\cdot)\ {\stackrel{v}{\rightarrow}}\ \mu(\!\cdot\!),\quad {n\to\infty},\end{equation}

holds on E, then

\begin{equation*}\widetilde N_p= \sum_{i=1}^{P_n}\varepsilon_{X_{ni}} \ {\stackrel{d}{\rightarrow}}\ N,\quad {n\to\infty},\end{equation*}

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

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

\begin{equation*}C_{K}^{+}(E)= \{ g \colon E \to {\mathbb{R}}_{+} \colon g \mbox{ continuous with compact support} \}.\end{equation*}

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

\begin{equation*}\mathbb{E}\bigg[ \exp\bigg({-}\int_E f\,{\text{d}} \widetilde N_p\bigg)\bigg] =\mathbb{E}\bigg[ \exp\bigg({-}\sum_{j=1}^{P_n}f(X_{nj}) \bigg)\bigg] =\mathbb{E} [ ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{P_n} ] .\end{equation*}

In view of (4.26) there is a real sequence $\epsilon_n\downarrow 0$ such that

(4.28) \begin{equation}\lim_{{n\to\infty}}\mathbb{P} (|P_n/p_n-1|>\epsilon_n )=\mathbb{P}(A_n^c)=0.\end{equation}

Then

\begin{align*}& \mathbb{E} [ ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{P_n} ]\\*&\quad = \mathbb{E} [ ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{P_n} (\textbf{1}(A_n^c)+\textbf{1}(A_n) ) ]\\*&\quad = I_1+I_2.\end{align*}

By (4.28) we have $I_1\le \mathbb{P}(A_n^c)\to 0$ as ${n\to\infty}$ , while

(4.29) \begin{equation} \mathbb{E} [ ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{(1+\epsilon_n)p_n}\textbf{1}(A_n) ] \le I_2\le \mathbb{E} [ ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{(1-\epsilon_n)p_n}\textbf{1}(A_n) ].\end{equation}

In view of Proposition 2.1 and (4.27)

\begin{equation*} ( \mathbb{E} [ \exp\!({-}f(X_{n1}) ] ) ^{(1\pm\epsilon_n)p_n}\to \mathbb{E}\bigg(\!\exp\bigg({-}\int_E (1-\text{e}^{-f(\textbf{x})})\,\mu({\text{d}} \textbf{x})\bigg).\end{equation*}

The right-hand side is the Laplace functional of a $\text{PRM}(\mu)$ . Now an application of dominated convergence to $I_2$ in (4.29) 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 (4.26). 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 i.i.d. 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

\begin{equation*}N_p= \frac{1 }{k} \sum_{i=1}^{p} \varepsilon_{c_{[p/k]}^{-1}(S_{ni}/\sqrt{n}-d_{[p/k]})}{,}\end{equation*}

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 4.9. Assume the conditions and notation of Theorem 4.1. In this case, choose $c_n=1/d_n$ . Then

\begin{align*}\mathbb{E}[N_p(x,\infty)]&= \frac{p}{k}\,\mathbb{P} ( S_n/ \sqrt{n}>d_{[p/k]}+x/d_{[p/k]} )\to \text{e}^{-x},\\*\text{var} (N_p(x,\infty) )&\le \frac{p}{k^2} \,\mathbb{P} ( S_n/ \sqrt{n}>d_{[p/k]}+x/d_{[p/k]} )\to 0,\quad x\in\mathbb{R}, \quad {n\to\infty},\end{align*}

provided $p/k<\exp\!(\gamma_n^2/2)$ . It is not difficult to see that

\begin{equation*}N_p \ {\stackrel{\mathbb{P}}{\rightarrow}}\ -\log \Lambda.\end{equation*}

Similarly, assume the conditions and the notation of Theorem 4.3 and consider

\begin{equation*}N_p= \frac{1 }{k} \sum_{i=1}^{p} \varepsilon_{a_{[np/k]}^{-1}(S_{ni}-d_n)}.\end{equation*}

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

\begin{align*}\mathbb{E}[N_p(x,\infty)]&= \frac{p}{k}\,\mathbb{P} (a_{[np/k]}^{-1}(S_n-d_n)>x )\sim \frac{np}{k}\,\mathbb{P}(X>a_{[np/k]}\, x) \to p_+\,x^{-\alpha}=\mu_\alpha(x,\infty),\\\text{var}(N_p(x,\infty))&\to 0,\\\mathbb{E}[N_p({-}\infty,-x]]&= \frac{p}{k}\,\mathbb{P} (a_{[np/k]}^{-1}(S_n-d_n)\le -x )\to p_-\,x^{-\alpha}=\mu_\alpha({-}\infty,-x],\\*\text{var}(N_p({-}\infty,-x])&\to 0,\end{align*}

provided the modified sequence $p_n/k_n\to\infty$ satisfies the conditions imposed on ${(p_n)}$ in Theorem 4.3. 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 $N_p \ {\stackrel{\mathbb{P}}{\rightarrow}}\ \mu_\alpha$ . Then, following the lines of Resnick and Stărică [Reference Resnick and Stărică30, 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 $S_{n,(1)}\ge\cdots\ge S_{n,(k)}$ for the k largest values. Then

\begin{equation*}\frac {1}{k} \sum_{i=1}^k \log \frac{S_{n,(i)}}{S_{n,(k)}} \ {\stackrel{\mathbb{P}}{\rightarrow}}\ \frac 1 \alpha.\end{equation*}

4.6. Some related results

The largest values of sequences of i.i.d. 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 i.i.d. 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

\begin{equation*}\textbf{X}\,:\!=\textbf{X}_n=(X_{it})_{i=1,\ldots,p;t=1,\ldots,n}\end{equation*}

and the corresponding sample covariance matrix $\textbf{X}\textbf{X}^\top=(S_{ij})$ . Heiny and Mikosch [Reference Heiny and Mikosch12] proved that

\begin{equation*} a_{np}^{-2}\|\textbf{X}\textbf{X}^\top - \operatorname{diag} (\textbf{X}\textbf{X}^\top)\|_2 \ {\stackrel{\mathbb{P}}{\rightarrow}}\ 0,\quad {n\to\infty},\end{equation*}

where $\|A\|_2$ denotes the spectral norm of a $p\times p$ symmetric matrix A, $\operatorname{diag} (A)$ 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 ${\lambda}_{(1)}(A)\ge \cdots\ge {\lambda}_{(p)}(A)$ for the ordered eigenvalues of A. According to Weyl’s inequality (see Bhatia [Reference Bhatia2]), the eigenvalues of $\textbf{X}\textbf{X}^\top$ satisfy the relation

(4.30) \begin{equation}a_{np}^{-2}\sup_{i=1,\ldots,p} |{\lambda}_{(i)}(\textbf{X}\textbf{X}^\top)- {\lambda}_{(i)}(\!\operatorname{diag} (\textbf{X}\textbf{X}^\top)) |\le a_{np}^{-2}\|\textbf{X}\textbf{X}^\top -\operatorname{diag} (\textbf{X}\textbf{X}^\top)\|_2 \ {\stackrel{\mathbb{P}}{\rightarrow}}\ 0.\end{equation}

But of course, ${\lambda}_{(i)}(\!\operatorname{diag} (\textbf{X}\textbf{X}^\top))$ are the ordered values of the i.i.d. partial sums $S_{ii}=\sum_{t=1}^n X_{it}^2$ , $i=1,\ldots,p$ . In view of (4.30) the asymptotic theory for the largest eigenvalues of the normalized sample covariance matrix $a_{np}^{-2} \textbf{X}\textbf{X}^\top$ (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, (4.30) 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 $\textbf{X}\textbf{X}^\top$ provided X has sufficiently high moments. Heiny, Mikosch, and Yslas [Reference Heiny, Mikosch and Yslas13] prove Gumbel convergence for the point process of the off-diagonal elements $(S_{ij})_{1\le i<j\le p}$ . 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, Heiny et al. [Reference Heiny, Mikosch and Yslas13] 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 $\text{PRM}({-}\log \Lambda)$ . This result indicates that the off-diagonal and diagonal entries of $\textbf{X}\textbf{X}^\top$ exhibit very similar extremal behavior. This is in stark contrast to the aforementioned results in [Reference Heiny and Mikosch12], where the diagonal entries have Fréchet extremal behavior.

Related results can also be found in Gantert and Höfelsauer [Reference Gantert and Höfelsauer11], who consider real-valued branching random walks and prove a large deviation principle for the position of the rightmost particle; see [Reference Gantert and Höfelsauer11, Theorem 3.2]. The position of the rightmost 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 random walks considered are i.i.d. They show that the maximum of these i.i.d. random walks stochastically dominates the maximum of the branching random walks; see [Reference Gantert and Höfelsauer11, Theorem 3.1 and Lemma 5.2]. An early comparison between maxima of branching and i.i.d. random walks was provided by Durrett [Reference Durrett7].