∎

¹¹institutetext: N. M. Markovich ²²institutetext: V.A.Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, 117997 Moscow, Russia
Tel.: +007-495-3348820
Fax: +007-495-3342016
²²email: [email protected]

WEIGHTED MAXIMA AND SUMS OF NON-STATIONARY RANDOM LENGTH SEQUENCES IN HEAVY-TAILED MODELS

Natalia M. Markovich

(Received: date / Accepted: date)

Abstract

The sums and maxima of weighted non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraints are that there exists a unique series in a scheme of series with the minimum tail index, the tail of the term number is lighter than the tail of the terms and the weights are positive constants. These assumptions are changed here: a bounded random number of series is allowed to have the minimum tail index, the tail of the term number may be heavier than the tail of the terms and the weights may be real-valued. Then we derive the tail and extremal indices of the weighted non-stationary random length sequences under the new assumptions.

Keywords:

Random length sequences non-stationarity regularly varying tail extremal index tail index real-valued weights

pacs:

60G70 62G32

1 Introduction

Random length sequences and distribution tails of their sums and maxima attract the interest of many researchers due to numerous applications including queues, branching processes and random networks, see Asmussen and Foss (2018), Jessen and Mikosch (2006), Goldaeva and Lebedev (2018), Lebedev (2015a, 2015b), Markovich and Rodionov (2020), Olvera-Cravioto (2012), Robert and Segers (2008), Tillier and Wintenberger (2018).

Considering weighted sums and maxima we aim to extend the results obtained in Markovich and Rodionov (2020). As in the latter paper, we deal with a doubly-indexed array $\{Y_{n,i}:n,i\geq 1\}$ of nonnegative random variables (r.v.s) in which the ”row index” $n$ corresponds to the time, and the ”column index” $i$ enumerates the series. On the same probability space, the existence of a sequence of non-negative integer-valued r.v.s $\{N_{n}:n\geq 1\}$ is assumed. Let $\{Y_{n,i}:n\geq 1\}$ be a strict-sense stationary sequence with the extremal index $\theta_{i}$ having a regularly varying tail

\displaystyle P\{Y_{n,i}>x\}

\displaystyle=

\displaystyle\ell_{i}(x)x^{-k_{i}}

(1)

with tail index $k_{i}>0$ and a slowly varying function $\ell_{i}(x)$ for any $i\geq 1$ . There are no assumptions on the dependence structure in $i$ .

Definition 1

A strictly stationary sequence $\{Y_{n}\}_{n\geq 1}$ with distribution function $F(x)$ and $M_{n}=\bigvee_{j=1}^{n}Y_{j}=\max_{j}Y_{j}$ is said to have the extremal index $\theta\in[0,1]$ if for each $0<\tau<\infty$ there exists a sequence of real numbers $u_{n}=u_{n}(\tau)$ such that

\displaystyle\lim_{n\to\infty}n(1-F(u_{n}))=\tau,

(2)

\lim_{n\to\infty}P\{M_{n}\leq u_{n}\}=e^{-\tau\theta}

(3)

hold (Leadbetter et al. (1983), p.63).

I.i.d. r.v.s $\{Y_{n}\}$ give $\theta=1$ . The converse may be incorrect. An extremal index that is close to zero implies a kind of a strong local dependence.

In Markovich and Rodionov (2020), the weighted sums and maxima

	$\displaystyle Y_{n}^{*}(z,N_{n})=\max(z_{1}Y_{n,1},...,z_{N_{n}}Y_{n,N_{n}}),$		(4)
	$\displaystyle~{}~{}Y_{n}(z,N_{n})=z_{1}Y_{n,1}+...+z_{N_{n}}Y_{n,N_{n}}$

for positive constants $z_{1},z_{2},...$ and regularly varying $\{Y_{n,i}:n\geq 1\}$ were considered. A similar result was obtained in Goldaeva (2013) for the same weights and for random sequences of a fixed length $l\geq 1$ and when $\{Y_{n,i}:n\geq 1\}$ have a power-type tail, i.e. $P(Y_{n,i}>x)\sim c^{(i)}x^{-k_{i}}$ as $x\to\infty$ ,¹¹1The symbol $\sim$ means asymptotically equal to or $f(x)\sim g(x)$ $\Leftrightarrow$ $f(x)/g(x)\rightarrow 1$ as $x\rightarrow a$ , $x\in M$ where the functions $f(x)$ and $g(x)$ are defined on some set $M$ and $a$ is a limit point of $M$ . where $c^{(i)}$ is a positive constant for each fixed $i$ .
Let us recall Theorem 4 derived in Markovich and Rodionov (2020) and $\{Y_{n,i}:n\geq 1\}$ related to sums and maxima (4) in the following Theorem 1.2. It is assumed that the ”column” sequences $\{Y_{n,i}:i\geq 1\}$ have stationary distribution tails (1) in $n$ with positive tail indices $\{k_{1},k_{2},...\}$ and extremal indices $\{\theta_{1},\theta_{2},...\}$ for each fixed $i$ . Here $\{\ell_{i}(x)\}$ are restricted by the condition: for all $A>1$ , $\delta>0$ there exists $x_{0}(A,\delta)$ such that for all $i\geq 1$

\displaystyle\ell_{i}(x)\leq Ax^{\delta},\ \ x>x_{0}(A,\delta)

(5)

holds. $N_{n}$ has a regularly varying distribution with the tail index $\alpha>0,$ that is

\displaystyle P(N_{n}>x)=x^{-\alpha}\tilde{\ell}_{n}(x).

(6)

There is a minimum tail index $k_{1}$ and $k:=\lim_{n\to\infty}\inf_{2\leq i\leq l_{n}}k_{i},$

\displaystyle l_{n}=[n^{\chi}],\qquad

(7)

and $\chi$ satisfies

0<\chi<\chi_{0},\qquad\chi_{0}=\frac{k-k_{1}}{k_{1}(k+1)}.

(8)

An arbitrary dependence structure between $\{Y_{n,i}\}$ and $\{N_{n}\}$ is allowed. The tail of $N_{n}$ does not dominate the tail of the most heavy-tailed term $Y_{n,1}$ . Let $\ell(x)$ be such that $\ell_{1}(x)=\ell^{-k_{1}}(x)$ and $\ell^{\sharp}(x)$ be the de Brujin conjugate of $\ell(x)$ (see Bingham et al. (1987)). Let

\displaystyle u_{n}

\displaystyle=

\displaystyle yn^{1/k_{1}}\ell_{1}^{\sharp}(n),~{}~{}y>0,

(9)

where we denote $\ell_{1}^{\sharp}(x)=\ell^{\sharp}(x^{1/k_{1}})$ and the positive weights $\{z_{i}\}$ are bounded.

Theorem 1.1

(Markovich and Rodionov 2020) Let $k_{1}<k$ , (5), (7) and (8) hold. Then the sequences $Y_{n}^{*}(z,l_{n})$ and $Y_{n}(z,l_{n})$ have the same tail index $k_{1}$ and the same extremal index $\theta_{1}$ .

Theorem 1.2 follows by Theorem 1.1.

Theorem 1.2

(Markovich and Rodionov 2020) Let the sets of slowly varying functions $\{\tilde{\ell}_{n}(x)\}_{n\geq 1}$ in (6) and $\{\ell_{i}(x)\}_{i\geq 1}$ in (1) satisfy the condition (5). Suppose that $k_{1}<k$ and

\displaystyle P\{N_{n}>l_{n}\}

\displaystyle=

\displaystyle o\left(P\{Y_{n,1}>u_{n}\}\right),~{}~{}n\to\infty

(10)

hold, where the sequence $l_{n}$ satisfies (7) and (8). Then the sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ have the same tail index $k_{1}$ and the same extremal index $\theta_{1}$ .

Remark 1

By the proof of the latter theorems it follows that if the ”column” series with a minimum tail index $k_{1}$ is unique, then the tail distributions of the sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ are asymptotically equivalent to its distribution tail. This is enough for $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ to have an extremal index $\theta_{1}$ , despite both sequences may be non-stationary distributed due to an arbitrary dependence between the ”column” sequences.

Our objectives are to revise Theorem 1.2 for the case when a random number of ”column” series may have the minimum tail index both for positive and real-valued constant weights in (4). Moreover, we observe how the results may change under the opposite assumption

\displaystyle P\{Y_{n,1}>u_{n}\}

\displaystyle=

\displaystyle o\left(P\{N_{n}>l_{n}\}\right),~{}~{}n\to\infty

(11)

instead of (10).
We use the following notations

$\displaystyle M_{n}^{(i)}$	$\displaystyle=$	$\displaystyle\max\{Y_{1,i},Y_{2,i},...,Y_{n,i}\},~{}i\in\{1,..,l_{n}\},$
$\displaystyle M_{n}(z,l_{n})$	$\displaystyle=$	$\displaystyle\max\{Y_{1}(z,l_{n}),Y_{2}(z,l_{n}),...,Y_{n}(z,l_{n})\},$
$\displaystyle M_{n}^{*}(z,l_{n})$	$\displaystyle=$	$\displaystyle\max\{Y_{1}^{}(z,l_{n}),...,Y_{n}^{}(z,l_{n})\}=\max\{z_{1}M_{n}^{(1)},...,z_{l_{n}}M_{n}^{(l_{n})}\},~{}~{}n\geq 1.$

The paper is organized as follows. The revision of Theorems 1.1 and 1.2 by Theorems 2.1 and 2.2 for fixed and random numbers of the most heavy-tailed ”column” series and positive weights in (4) is given in Section 2.1. The same revision for real-valued weights is given in Section 2.2. The revision of Theorem 1.2 for a heavy-tailed number $N_{n}$ of light-tailed terms is stated in Section 2.3. The proofs are given in Section 3.

2 Main Results

2.1 Revision of Theorem 1.2 for positive constant weights

We revise Theorem 1.2 by Theorem 2.2 allowing a random bounded number $d\geq 1$ of series to have a minimum tail index. To this end, we extend Theorem 1.1 by Theorem 2.1. Theorem 1.1 covers the case $d=1$ . We assume in Theorem 2.1 that $d>1$ is fixed and $k_{i}=k_{1}$ , $i\in\{1,...,d\}$ , $1\leq d\leq l_{n}-1$ , $k_{1}<k=k_{d+1}$ , where

\displaystyle k

\displaystyle:=

\displaystyle\lim_{n\to\infty}\inf_{d+1\leq i\leq l_{n}}k_{i}

(12)

holds. We introduce the following conditions:

(A1)

The stationary sequences $\{Y_{n,i}\}_{n\geq 1}$ , $i\in\{1,...,d\}$ are mutually independent, and independent of the sequences $\{Y_{n,i}\}_{n\geq 1}$ , $i\in\{d+1,...,l_{n}\}$ .

(A2)

Assume $\{Y_{n,i}\}_{n\geq 1}$ , $i\in\{1,...,d\}$ satisfy the following conditions as $x\to\infty$

\displaystyle\frac{P\{Y_{n,i}>x\}}{x^{-k_{1}}\ell_{1}(x)}

\displaystyle\rightarrow

\displaystyle c_{i},~{}~{}i\in\{1,...,d\},

(13)

for some non-negative numbers $c_{i}$ ,

\displaystyle\frac{P\{Y_{n,i}>x,Y_{n,j}>x\}}{x^{-k_{1}}\ell_{1}(x)}

\displaystyle\rightarrow

\displaystyle 0,~{}~{}i\neq j,~{}~{}i,j\in\{1,...,d\}.

(14)

By Lemma 2.1 in Davis and Resnik (1996) in conditions (A2) it holds

\displaystyle\frac{P\{\sum_{i=1}^{d}Y_{n,i}>x\}}{x^{-k_{1}}\ell_{1}(x)}

\displaystyle\rightarrow

\displaystyle\sum_{i=1}^{d}c_{i},~{}~{}x\to\infty

(15)

for any $n\geq 1$ .

(A3)

Assume that for each $n\geq 1$ there exists $i\in\{1,...,d\}$ such that

\displaystyle P\{\max_{1\leq j\leq d,j\neq i}(z_{j}Y_{n,j})>x,z_{i}Y_{n,i}\leq x\}=o(P\{z_{i}Y_{n,i}>x\}),~{}~{}x\to\infty

(16)

holds.

(A4)

Assume that there exists $i\in\{1,...,d\}$ such that it holds

\displaystyle P\{\max_{1\leq j\leq d,j\neq i}(z_{j}M_{n}^{(j)})>u_{n},z_{i}M_{n}^{(i)}\leq u_{n}\}=o(1),~{}~{}n\to\infty.

(17)

Example 1

Let $z_{1}Y_{n,1}\geq z_{2}Y_{n,2}\geq...\geq z_{d}Y_{n,d}$ for a given $n\geq 1$ hold. If $z_{1}Y_{n,1}\leq x$ holds, then $z_{i}Y_{n,i}\leq x$ for all $i\in\{2,3,...,d\}$ and $P\{\max_{2\leq j\leq d}(z_{j}Y_{n,j})>x,z_{1}Y_{n,1}\leq x\}=0$ follows. Since for any ”row” sequence $\{z_{i}Y_{n,i}:i\geq 1\}$ there is a maximum term, the condition (16) follows.

Example 2

The assumption (17) is particularly valid for the following $d$ ”column” sequences. Let elements of each ”column” sequence be sums of the corresponding elements of all previous ”columns”, i.e. $Y_{n,i}=\sum_{j=1}^{i-1}Y_{n,j}$ and $z_{1}\leq z_{2}\leq...\leq z_{d}$ . Each of the $d$ ”column” sequences has the tail index $k_{1}$ that follows from the statement of Theorem 2.1. Since $M_{n}^{(1)}=M_{n}^{(2)}<M_{n}^{(3)}<...<M_{n}^{(d)}$ holds, then $\sum_{j=1}^{d-1}P\{z_{j}M_{n}^{(j)}>u_{n},z_{j+1}M_{n}^{(j+1)}\leq u_{n},...,z_{d}M_{n}^{(d)}\leq u_{n}\}=0$ or equivalently (17) for $i=d$ follow. In the same way, (17) is valid for all $d$ ”column” sequences such that $M_{n}^{(1)}\leq M_{n}^{(2)}\leq M_{n}^{(3)}\leq...\leq M_{n}^{(d)}$ holds.

Let $u_{n}$ in Theorem 2.1 be as in (9).

Theorem 2.1

Let (5) hold for all $d+1\leq i\leq l_{n}$ , (7), (8) be satisfied. If, in addition, (A1) or (A2) holds, then $Y_{n}^{*}(z,l_{n})$ and $Y_{n}(z,l_{n})$ have tail index $k_{1}$ . If, instead of (A1) or (A2), (A3) holds, then $Y_{n}^{*}(z,l_{n})$ has the same tail index.

1.

If (A1) holds, then $Y_{n}^{*}(z,l_{n})$ and $Y_{n}(z,l_{n})$ have the same extremal index

$\displaystyle\theta(z)$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{d}\theta_{j}z_{j}^{k_{1}}/\sum_{j=1}^{d}z_{j}^{k_{1}}.$ (18)
2.

If (A4) holds, then $Y_{n}^{*}(z,l_{n})$ has the extremal index $\theta_{i}$ . If, additionally to (A4), (A2) holds, then $Y_{n}(z,l_{n})$ has the same extremal index as $Y_{n}^{*}(z,l_{n})$ .

Remark 2

An arbitrary dependence among elements of the $d$ ”columns” series with the minimum tail index in Item 2 of Theorem 2.1 (for instance, elements of odd ”row” sequences coincide and elements of the even rows are i.i.d.) leads to the non-stationary sequences of maxima $Y_{n}^{*}(z,l_{n})$ and sums $Y_{n}(z,l_{n})$ . However, the extremal index of $Y_{n}^{*}(z,l_{n})$ exists in case of (17) due to the asymptotic equivalence of the distributions of $M_{n}^{*}(z,d)$ and $M_{n}^{(i)}$ .

Now we reformulate Theorem 1.2.

Theorem 2.2

Let the sets of slowly varying functions $\{\ell_{i}(x)\}_{d+1\leq i\leq l_{n}}$ in (1) and $\{\tilde{\ell}_{n}(x)\}_{n\geq 1}$ in (6) satisfy the condition (5) and (7), (8), (10) hold. Assume that $d$ and $\{Y_{n,i}\}$ are independent.

(i)
Let $d$ be a bounded discrete r.v. such that $1<d<d_{n}=\min(C,l_{n})$ , $C>1$ holds.
1. (a)
  
  If (A1) or (A2) for any $d\in\{2,3,...,\lfloor d_{n}-1\rfloor\}$ holds and $N_{n}$ and $\{Y_{n,i}\}$ are independent, then $Y_{n}(z,N_{n})$ and $Y^{*}_{n}(z,N_{n})$ have the tail index $k_{1}$ . If, instead of (A1) and (A2), (A3) holds, then $Y_{n}^{*}(z,N_{n})$ has the same tail index.
2. (b)
  
  If (A4) where in (17) $d$ is replaced by $\lfloor d_{n}-1\rfloor$ holds, then $Y_{n}^{*}(z,N_{n})$ has the extremal index $\theta_{i}$ . If, in addition, (A1) (or (A2)) for any $d\in\{2,3,...,\lfloor d_{n}-1\rfloor\}$ holds, then $Y_{n}(z,N_{n})$ has the same extremal index.
(ii)

Suppose that $d>1$ is a bounded discrete r.v. equal to a positive integer a.s.. Then all statements of Item (i) are fulfilled.

2.2 Revision of Theorem 1.2 for real-valued constant weights

Let us consider (4) for real-valued constant weights. The sums and maxima may be partitioned into negative and positive parts

	$\displaystyle Y_{n}^{}(z,N_{n}^{+},N_{n}^{-})=\max(Y_{n}^{+}(z^{+},N_{n}^{+}),Y_{n}^{-}(z^{-},N_{n}^{-}))=Y_{n}^{+}(z^{+},N_{n}^{+}),$
	$\displaystyle Y_{n}^{*}(z,N_{n}^{+},N_{n}^{-})=\min(Y_{n}^{+}(z^{+},N_{n}^{+}),Y_{n}^{-}(z^{-},N_{n}^{-}))=Y_{n}^{-}(z^{-},N_{n}^{-}),$
	$\displaystyle~{}~{}Y_{n}(z,N_{n}^{+},N_{n}^{-})=Y_{n}^{+}(z^{+},N_{n}^{+})+Y_{n}^{-}(z^{-},N_{n}^{-}),$		(19)

where the terms marked by ’ $+$ ’ and ’ $-$ ’ correspond to sums and maxima with positive and negative weights and the corresponding random numbers of terms $N_{n}^{+}$ and $N_{n}^{-}$ . $N_{n}^{\pm}$ satisfies (6). For brevity we denote in the section $Y_{n}^{*}(z,N_{n}^{+},N_{n}^{-})$ , $Y_{n}^{**}(z,N_{n}^{+},N_{n}^{-})$ and $Y_{n}(z,N_{n}^{+},N_{n}^{-})$ as $Y_{n}^{*}(z,N_{n})$ , $Y_{n}^{**}(z,N_{n})$ and $Y_{n}(z,N_{n})$ , respectively.
We avoid to use only positive weights, the case derived in Markovich and Rodionov (2020), or only negative weights since then $P\{Y_{n}^{*}(z,N_{n})>u_{n}\}=P\{Y_{n}^{*}(z^{-},N_{n}^{-})>u_{n}\}=0$ and $P\{Y_{n}(z,N_{n})>u_{n}\}=P\{Y_{n}^{-}(z^{-},N_{n}^{-})>u_{n}\}=0$ hold for positive $u_{n}$ .
We assume that the number of column subsequences $l_{n^{+}}$ and $l_{n^{-}}$ of $l_{n}$ such that $l_{n^{+}}+l_{n^{-}}=l_{n}$ , $n^{\pm}\in\mathbb{N}$ , $n^{\pm}\to\infty$ for positive and negative weights, respectively, satisfy conditions similar to (7) and (8), i.e.

\displaystyle l_{n^{\pm}}=[{n^{\pm}}^{\chi^{\pm}}],\qquad

(20)

and $\chi^{\pm}$ satisfies

0<\chi^{\pm}<\chi_{0}^{\pm},\qquad\chi_{0}^{\pm}=\frac{k^{\pm}-k_{1}^{\pm}}{k_{1}^{\pm}(k^{\pm}+1)},

(21)

where

\displaystyle k^{+}:=\lim_{n\to\infty}\inf_{d^{+}+1\leq i\leq l_{n^{+}}}k_{i}^{+},\qquad k^{-}:=\lim_{n\to\infty}\inf_{d^{-}+1\leq i\leq l_{n^{-}}}k_{i}^{-},

(22)

Without loss of generality, we assume that the first $d^{+}$ and $d^{-}$ , $1\leq d^{\pm}\leq l_{n^{\pm}}-1$ , series in the corresponding subsequences have the minimum tail indices $k_{1}^{+}$ and $k_{1}^{-}$ , $k_{1}^{\pm}<k^{\pm}$ , respectively.
By (2.2) all statements of Theorems 1.2 and 2.2 regarding the maximum $Y_{n}^{*}(z,N_{n})$ (or $-Y_{n}^{**}(z,N_{n})$ ) are fulfilled with a replacement of $k_{1}$ by the minimum tail index $k_{1}^{+}$ (or $k_{1}^{-}$ ) of the positive (or negative) weighted column sequences subject to the required assumptions.

2.2.1 A unique ”column” series with a minimum tail index

Let there be a unique column with a positive weight which has a minimum tail index $k_{1}^{+}$ and the extremal index $\theta_{1}^{+}$ , and similarly, there be a unique column with a negative weight which has a minimum tail index $k_{1}^{-}$ and the extremal index $\theta_{1}^{-}$ . We select $u_{n}$ similar as in Theorem 1.2, i.e. $u_{n}^{+}=yn^{1/k_{1}^{+}}\ell^{\sharp}_{1}(n)$ or $u_{n}^{-}=yn^{1/k_{1}^{-}}\ell^{\sharp}_{1}(n)$ for $y>0$ . Let us denote the column series with the minimum tail indices $k_{1}^{+}$ and $k_{1}^{-}$ as $Y_{n,1}^{+}$ and $Y_{n,1}^{-}$ , respectively.

Corollary 1

Let the sets of slowly varying functions $\{\ell_{i}(x)\}_{i\geq 1}$ in (1) and $\{\tilde{\ell}_{n}(x)\}_{n\geq 1}$ in (6) satisfy the condition (5) and $k_{1}^{\pm}<k^{\pm}$ hold. Suppose that

\displaystyle P\{N_{n}^{\pm}>l_{n^{\pm}}\}

\displaystyle=

\displaystyle o\left(P\{Y_{n,1}^{\pm}>u_{n}^{\pm}\}\right),~{}~{}n\to\infty,

(23)

holds, where the subsequences $l_{n^{\pm}}$ satisfy (20) and (21).

1.

If $k_{1}^{+}\leq k_{1}^{-}$ holds, then the sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ have the same tail index $k_{1}^{+}$ and the same extremal index $\theta_{1}^{+}$ for $u_{n}^{+}$ , and their extremal indices do not exist for $u_{n}^{-}$ .
2.

If $k_{1}^{-}\leq k_{1}^{+}$ holds, then the sequences $-Y_{n}^{**}(z,N_{n})$ and $-Y_{n}(z,N_{n})$ have the same tail index $k_{1}^{-}$ and the same extremal index $\theta_{1}^{-}$ for $u_{n}^{-}$ , and their extremal indices do not exist for $u_{n}^{+}$ .

2.2.2 A random number of ”column” series with a minimum tail index

Let a random number of column series have a minimum tail index.

Corollary 2

Let the conditions of Theorem 2.2 with (23) instead of (10) be fulfilled, $d^{+}$ and $d^{-}$ be bounded discrete r.v.s such that $d^{\pm}<d_{n}^{\pm}=\min(C^{\pm},l_{n}^{\pm})$ , $C^{\pm}>1$ hold and $d^{+}$ and $d^{-}$ be independent of $\{Y_{n,i}\}$ .

1.

If $k_{1}^{+}\leq k_{1}^{-}$ holds, then all statements of Theorem 2.2 are fulfilled for sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ for $u_{n}^{+}$ , and the extremal index of $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ does not exist for $u_{n}^{-}$ .
2.

If $k_{1}^{-}\leq k_{1}^{+}$ holds, then all statements of the previous case are fulfilled by substituting $Y_{n}(z,N_{n})$ by $-Y_{n}(z,N_{n})$ , $Y_{n}^{*}(z,N_{n})$ by $-Y_{n}^{**}(z,N_{n})$ , $k_{1}^{+}$ by $k_{1}^{-}$ and $u_{n}^{\pm}$ by $u_{n}^{\mp}$ symmetrically.

2.3 Revision of Theorem 1.2 for a heavy-tailed number of light-tailed terms

The assumption (10) is crucial for the proof of Theorem 1.2. We will replace the latter assumption with an opposite one and derive new statements for a unique column sequence and a random number of column sequences with a minimum tail index for positive weights.

2.3.1 A unique ”column” series with a minimum tail index

Theorem 2.3

Let the conditions of Theorem 1.2 be fulfilled apart of (10) and the sequence $l_{n}$ satisfy (7) and (8).
(i) Assume (11) and

\displaystyle\alpha\chi_{0}

\displaystyle>

\displaystyle 1+\frac{\alpha}{k_{1}}\delta^{*},

(24)

for $\delta^{*}>0$ hold, where $\alpha>0$ be the tail index of $N_{n}$ . Then the sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ have the same tail index $k_{1}$ and if

\displaystyle\alpha\chi_{0}

\displaystyle>

\displaystyle 1+\frac{1}{2}(1-\alpha\chi),

(25)

holds, then the same extremal index $\theta_{1}$ .
(ii) Assume

\displaystyle P\{N_{n}>l_{n}\}

\displaystyle=

\displaystyle P\{z_{1}Y_{n,1}>u_{n}\}(1+o(1))

(26)

holds, and $N_{n}$ is regularly varying (6). Then $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ have the same tail index $k_{1}$ and the extremal index $\theta_{1}$ .

2.3.2 A random number of ”column” series with a minimum tail index

Theorem 2.4

Let the conditions of Theorem 2.2 be fulfilled, but (10) is substituted by (11) and (24). Then all statements of Theorem 2.2 regarding tail and extremal indices are fulfilled.

3 Proofs

3.1 Proof of Theorem 2.1

The proof extends Theorem 3 in Markovich and Rodionov (2020). We just indicate the modifications. The numeration of Theorem 3 in Markovich and Rodionov (2020) is preserved throughout the proof. For brevity, we denote in this section $Y_{n}(z)=Y_{n}(z,l_{n})$ , $Y_{n}^{*}(z)=Y_{n}^{*}(z,l_{n})$ , $M_{n}(z)=M_{n}(z,l_{n})$ and $M_{n}^{*}(z)=M_{n}^{*}(z,l_{n})$ .
The right-hand side of (12) in Markovich and Rodionov (2020) can be rewritten as

\displaystyle\!\!\!\!P\{Y_{n}(z)>u_{n}\}

\displaystyle\leq

\displaystyle P\{\sum_{i=1}^{d}z_{i}Y_{n,i}>u_{n}(1-\varepsilon)\}+\sum_{i=d+1}^{l_{n}}P\{z_{i}Y_{n,i}>u_{n}\varepsilon_{i}\},

(27)

where $\sum_{i=1}^{l_{n}}\varepsilon_{i}=1$ holds and $\{\varepsilon_{i}\}$ is a sequence of positive elements. Let us denote $\varepsilon=\sum_{i=d+1}^{l_{n}}\varepsilon_{i}$ . One may take $\varepsilon_{i}$ , $i\in\{d+1,...,l_{n}\}$ , in such a way to satisfy $\varepsilon_{i}\to 0$ and $\varepsilon\to 0$ as $n\to\infty$ . Choosing $\{\varepsilon_{i}=1/l_{n}^{\eta+1}\}$ , $\eta>0$ as in Lemma 1 in Markovich and Rodionov (2020) one can derive (13) in Markovich and Rodionov (2020) substituting $2$ by $d+1$ , namely, it holds

\displaystyle\sum_{i=d+1}^{l_{n}}P\{z_{i}Y_{n,i}>u_{n}\varepsilon_{i}\}=o(1/n),\qquad n\to\infty.

(28)

To prove the latter, we need to assume (5) for $d+1\leq i\leq l_{n}$ . For the selected $u_{n}$ we have

\displaystyle P\{z_{1}Y_{n,1}>u_{n}\}=(z_{1}/y)^{k_{1}}n^{-1}(1+o(1)),~{}~{}n\to\infty.

(29)

We obtain

			$\displaystyle P\{\sum_{i=1}^{d}z_{i}Y_{n,i}>u_{n}(1-\varepsilon)\}\leq\sum_{i=1}^{d}P\{z_{i}Y_{n,i}>u_{n}(1-\varepsilon)\varepsilon_{i}^{*}\}$		(30)
		$\displaystyle=$	$\displaystyle\frac{n^{-1}}{(y(1-\varepsilon))^{k_{1}}}\sum_{i=1}^{d}\left(\frac{z_{i}}{\varepsilon_{i}^{}}\right)^{k_{1}}(1+o(1))=\left(\frac{z^{}}{y(1-\varepsilon)}\right)^{k_{1}}n^{-1}(1+o(1)),$		(30)

where $\sum_{i=1}^{d}\varepsilon_{i}^{*}=1$ and $(z^{*})^{k_{1}}=\sum_{i=1}^{d}\left(z_{i}/\varepsilon_{i}^{*}\right)^{k_{1}}$ hold. By (27)-(30) it follows

			$\displaystyle\left(z_{1}/y\right)^{k_{1}}n^{-1}(1+o(1))=P\{z_{1}Y_{n,1}>u_{n}\}\leq P\{Y_{n}^{*}(z)>u_{n}\}$		(31)
		$\displaystyle\leq$	$\displaystyle P\{Y_{n}(z)>u_{n}\}\leq\left(z^{*}/y(1-\varepsilon)\right)^{k_{1}}n^{-1}(1+o(1))+o(1/n).$		(31)

If the condition (A2) holds and replacing $x$ in (15) by $u_{n}$ , we get for any $m\geq 1$

\displaystyle P\{\sum_{i=1}^{d}z_{i}Y_{m,i}>u_{n}\}

\displaystyle=

\displaystyle n^{-1}\sum_{i=1}^{d}\left(\frac{z_{i}}{y}\right)^{k_{1}}c_{i}(1+o(1))

since by (13) and (29)

\displaystyle\frac{P\{z_{i}Y_{n,i}>u_{n}\}}{P\{z_{1}Y_{n,1}>u_{n}\}}

\displaystyle=

\displaystyle\left(\frac{z_{i}}{z_{1}}\right)^{k_{1}}c_{i}(1+o(1)),~{}~{}n\to\infty,~{}~{}i=1,2,...,d.

By (27) and (28) this implies

\displaystyle\lim_{n\to\infty}nP\{Y_{n}(z)>u_{n}\}

\displaystyle=

\displaystyle\lim_{n\to\infty}nP\{Y_{n}^{*}(z)>u_{n}\}=\sum_{i=1}^{d}\left(\frac{z_{i}}{y}\right)^{k_{1}}c_{i}.

(32)

The latter is valid for $Y_{m}^{*}(z)$ by (13), (14) and due to

			$\displaystyle P\{\max(z_{1}Y_{m,1},z_{2}Y_{m,2})>u_{n}\}$
		$\displaystyle=$	$\displaystyle P\{z_{1}Y_{m,1}>u_{n}\}+P\{z_{2}Y_{m,2}>u_{n}\}-P\{z_{1}Y_{m,1}>u_{n},z_{2}Y_{m,2})>u_{n}\}$

(the case for general $d$ follows by induction) and since it holds

\displaystyle P\{Y_{n}^{*}(z,d)>u_{n}\}\leq P\{Y_{n}^{*}(z)>u_{n}\}\leq P\{Y_{n}(z,d)>u_{n}\}.

Tail index

Let us find the tail index of $Y_{m}(z)$ and $Y_{m}^{*}(z)$ for all $m\geq 1$ . Notice that the relation (27) remains true when replacing $u_{n}$ by $x>0$ . Similarly to (30) it holds

	$\displaystyle P\{\sum_{i=1}^{d}z_{i}Y_{m,i}>x(1-\varepsilon)\}$	$\displaystyle\leq$	$\displaystyle\frac{1}{(x(1-\varepsilon))^{k_{1}}}\sum_{i=1}^{d}\left(\frac{z_{i}}{\varepsilon_{i}^{*}}\right)^{k_{1}}\ell_{i}(x)(1+o(1))$
		$\displaystyle=$	$\displaystyle\frac{\ell^{*}(x)}{x^{k_{1}}}(1+o(1)),~{}~{}x\to\infty,$

where $\ell^{*}(x)=\sum_{i=1}^{d}\left(z_{i}/(\varepsilon_{i}^{*}(1-\varepsilon))\right)^{k_{1}}\ell_{i}(x)$ is a slowly varying function. Formula (16) in Markovich and Rodionov (2020) is fulfilled by replacing $2$ by $d+1$ in the sum, namely,

\displaystyle\sum_{i=d+1}^{l_{m}}P\{z_{i}Y_{n,i}>x\varepsilon_{i}\}

\displaystyle\leq

\displaystyle O\left(x^{-k_{1}(1+\delta)}\right),~{}~{}x\to\infty,

(33)

where $\delta\in(0,(k-k_{1})/(k+1))$ . Then it follows that

	$\displaystyle(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))=P\{Y_{m,1}(z)>x/z_{1}\}\leq P\{Y^{*}_{m}(z)>x\}$	(34)
$\displaystyle\leq$	$\displaystyle P\{Y_{m}(z)>x\}\leq P\{\sum_{i=1}^{d}z_{i}Y_{m,i}>x(1-\varepsilon)\}+\sum_{i=d+1}^{l_{m}}P\{z_{i}Y_{n,i}>x\varepsilon_{i}\}$
$\displaystyle\leq$	$\displaystyle\ell^{*}(x)x^{-k_{1}}(1+o(1)).$

By (34) it does not follow that $Y_{m}(z)$ and $Y_{m}^{*}(z)$ are regularly varying.

Condition (A1)

The tail index of $Y_{n}(z)$ and $Y_{n}^{*}(z)$ is equal to $k_{1}$ if the condition (A1) holds. Really, it follows

\displaystyle P\{Y_{n}^{*}(z)>x\}

\displaystyle=

\displaystyle\sum_{i=1}^{d}\left(\frac{z_{i}}{x}\right)^{k_{1}}\ell_{i}(x)(1+o(1)),~{}~{}x\to\infty

(35)

by (40) with replacement $u_{n}$ by $x>0$ , and

\displaystyle\!\!\!\!\!P\{Y_{n}(z)>x\}

\displaystyle=

\displaystyle\sum_{i=1}^{d}P\{z_{i}Y_{i}>x\}(1+o(1))=\sum_{i=1}^{d}\left(\frac{z_{i}}{x}\right)^{k_{1}}\ell_{i}(x)(1+o(1))

(36)

as $x\to\infty$ by Lemma 3.1 in Jessen and Mikosch (2006) due to independence of the first $d$ ”column” sequences.

Conditions (A2)

In conditions (A2) and due to (15), (33), (34) we get

\displaystyle P\{Y_{m}(z)>x\}=x^{-k_{1}}\ell_{1}(x)\sum_{i=1}^{d}z_{i}^{k_{1}}c_{i}(1+o(1))

(37)

since $P\{Y_{m}(z)>x\}\geq P\{\sum_{i=1}^{d}z_{i}Y_{m,i}>x\}$ . The same is valid for $Y_{m}^{*}(z)$ by (13) and (14) in the same way as above. Hence, $Y_{m}(z)$ and $Y_{m}^{*}(z)$ have the same tail index $k_{1}$ for arbitrary (fixed) $m\geq 1$ .

Condition (A3)

We have for all $m\geq 2$

\displaystyle P\{Y_{m}^{*}(z)>x\}=P\{z_{1}Y_{m,1}>x\}+\sum_{i=2}^{l_{m}}P\{z_{i}Y_{m,i}>x,\max_{1\leq j\leq i-1}z_{j}Y_{m,j}\leq x\}.

Let us denote

\displaystyle P\{z_{i}Y_{m,i}>x,\max_{1\leq j\leq i-1}z_{j}Y_{m,j}\leq x\}

\displaystyle=

\displaystyle\xi_{i}.

(38)

We obtain

\displaystyle\sum_{i=2}^{l_{m}}\xi_{i}

\displaystyle=

\displaystyle\sum_{i=2}^{d}\xi_{i}+\sum_{i=d+1}^{l_{m}}\xi_{i}.

By (33) we have

\displaystyle\sum_{i=d+1}^{l_{m}}\xi_{i}\leq\sum_{i=d+1}^{l_{m}}P\{z_{i}Y_{m,i}>x\}\leq O\left(x^{-k_{1}(1+\delta)}\right),~{}x\to\infty.

By (16) it holds

	$\displaystyle\sum_{i=2}^{d}\xi_{i}$	$\displaystyle=$	$\displaystyle\sum_{i=2}^{d}P\{z_{i}Y_{m,i}>x,\max_{1\leq j\leq i-1}z_{j}Y_{m,j}\leq x\}$
		$\displaystyle=$	$\displaystyle P\{\max_{2\leq j\leq d}(z_{j}Y_{m,j})>x,z_{1}Y_{m,1}\leq x\}=o(P\{z_{1}Y_{m,1}\leq x\}),~{}~{}x\to\infty.$

Then it follows

\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!P\{Y^{*}_{m}(z))>x\}=P\{z_{1}Y_{m,1}>x\}(1+o(1))=(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))

and $Y^{*}_{m}(z)$ has tail index $k_{1}$ .

The extremal index for $d$ independent ”column” sequences

Let us assume that the condition (A1) hold. Due to independence we have

\displaystyle P\{M_{n}^{*}(z)\leq u_{n}\}=\prod_{j=1}^{d}P\{z_{j}M_{n}^{(j)}\leq u_{n}\}P\{z_{d+1}M_{n}^{(d+1)}\leq u_{n},...,z_{l_{n}}M_{n}^{(l_{n})}\leq u_{n}\}.

Since the extremal index of the sequence $\{Y_{n,j}\}_{n\geq 1}$ , $1\leq j\leq d$ is assumed to be equal to $\theta_{j},$ by (3) and (29) we get

\displaystyle\lim_{n\to\infty}\prod_{j=1}^{d}P\{z_{j}M_{n}^{(j)}\leq u_{n}\}

\displaystyle=

\displaystyle\exp(-\sum_{j=1}^{d}\theta_{j}(z_{j}/y)^{k_{1}}).

By (19) and (21) in Markovich and Rodionov (2020) where $d+2$ replaces $2$ when indexing the sums and by (3), we obtain

\displaystyle P\{z_{d+1}M_{n}^{(d+1)}\leq u_{n},...,z_{l_{n}}M_{n}^{(l_{n})}\leq u_{n}\}=\!P\{z_{d+1}M_{n}^{(d+1)}\leq u_{n}\}(1+o(1))\to 1

assuming $k=k_{d+1}$ , since

\displaystyle nP\{z_{d+1}Y_{n,d+1}>u_{n}\}=n\cdot n^{-k_{d+1}/k_{1}}(z_{d+1}/y)^{k_{d+1}}\ell(n)(1+o(1))\to 0

as $n\to\infty$ due to $k_{d+1}>k_{1}$ holds. We obtain

\displaystyle\lim_{n\to\infty}nP\{Y_{n}^{*}(z)>u_{n}\}=\sum_{j=1}^{d}\left(\frac{z_{j}}{y}\right)^{k_{1}}

(39)

since it holds

			$\displaystyle P\{Y_{n}^{*}(z)>u_{n}\}=P\{\max(z_{1}Y_{n,1},...,z_{d}Y_{n,d},z_{d+1}Y_{n,d+1},...,z_{l_{n}}Y_{n,l_{n}})>u_{n}\}$
		$\displaystyle=$	$\displaystyle 1-\prod_{i=1}^{d}(1-P\{z_{i}Y_{n,i}>u_{n}\})(1-P\{\max(z_{d+1}Y_{n,d+1},...,z_{l_{n}}Y_{n,l_{n}})>u_{n}\}).$

			$\displaystyle\prod_{i=1}^{d}(1-P\{z_{i}Y_{n,i}>u_{n}\})=1-\sum_{i=1}^{d}P\{z_{i}Y_{n,i}>u_{n}\}$
		$\displaystyle+$	$\displaystyle\sum_{i<j}P\{z_{i}Y_{n,i}>u_{n}\}P\{z_{j}Y_{n,j}>u_{n}\}-...-(-1)^{d-1}\prod_{i=1}^{d}P\{z_{i}Y_{n,i}>u_{n}\}$
		$\displaystyle=$	$\displaystyle\left(1-\sum_{i=1}^{d}P\{z_{i}Y_{n,i}>u_{n}\}\right)(1+o(1)),~{}~{}n\to\infty$

and by (29) it follows

	$\displaystyle P\{Y_{n}^{*}(z)>u_{n}\}$	(40)
$\displaystyle=$	$\displaystyle\!\!\!\left(\sum_{i=1}^{d}P\{z_{i}Y_{n,i}>u_{n}\}+P\{\max(z_{d+1}Y_{n,d+1},...,z_{l_{n}}Y_{n,l_{n}})>u_{n}\}\right)\cdot(1+o(1))$
$\displaystyle=$	$\displaystyle\sum_{i=1}^{d}\left(\frac{z_{i}}{y}\right)^{k_{1}}\cdot n^{-1}(1+o(1))+o(1/n)$

due to (9), (12), (13) and Lemma 1 in Markovich and Rodionov (2020) and $k=k_{d+1}$ . By Lemma 3.1 in Jessen and Mikosch (2006) it follows

\displaystyle\lim_{n\to\infty}nP\{Y_{n}(z)>u_{n}\}=\sum_{i=1}^{d}\left(\frac{z_{i}}{y}\right)^{k_{1}}.

(41)

By (39) and

\displaystyle P\{M_{n}^{*}(z)\leq u_{n}\}=\exp(-\sum_{j=1}^{d}\theta_{j}(z_{j}/y)^{k_{1}})(1+o(1))

we obtain that the extremal index of $Y_{n}^{*}(z)$ is equal to (18). In the same way as in the proof of Theorem 3 in Markovich and Rodionov (2020) one can show that $Y_{n}(z)$ has the same extremal index. Really, we obtain

	$\displaystyle 0\leq P\{M_{n}^{*}(z)\leq u_{n}\}-P\{M_{n}(z)\leq u_{n}\}$	(42)
$\displaystyle=$	$\displaystyle\sum_{j=1}^{n-1}P\{M_{n}^{*}(z)\leq u_{n},Y_{j}(z)>u_{n},Y_{j+1}(z)\leq u_{n},...,Y_{n}(z)\leq u_{n}\}$
$\displaystyle+$	$\displaystyle P\{M_{n}^{*}(z)\leq u_{n},Y_{n}(z)>u_{n}\}$
$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}P\{Y_{j}^{*}(z)\leq u_{n},Y_{j}(z)>u_{n}\}.$

It follows

\displaystyle\!\!\!\!\!\!\!\!\!\!P\{Y_{j}^{*}(z)\leq u_{n},Y_{j}(z)>u_{n}\}

\displaystyle=

\displaystyle P\{Y_{j}(z)>u_{n}\}-P\{Y_{j}^{*}(z)>u_{n}\}=o(n^{-1})

(43)

by (40) and (41) for $j\leq n$ .

The extremal index for $d$ dependent ”column” sequences

We find the extremal index of the sequences $Y_{n}^{*}(z)$ and $Y_{n}(z)$ if there is the dependence of $\{Y_{n,i}\}_{n\geq 1}$ in $i$ .
Let us rewrite (19) in Markovich and Rodionov (2020) as

	$\displaystyle P\{M_{n}^{*}(z,d)\leq u_{n}\}-\sum_{i=d+1}^{l_{n}}P\{z_{i}M_{n}^{(i)}>u_{n}\}$	(44)
$\displaystyle\leq$	$\displaystyle P\{z_{1}M_{n}^{(1)}\leq u_{n},...,z_{l_{n}}M_{n}^{(l_{n})}\leq u_{n}\}=P\{M_{n}^{*}(z)\leq u_{n}\}$
$\displaystyle\leq$	$\displaystyle P\{M_{n}^{*}(z,d)\leq u_{n}\}\leq P\{z_{d}M_{n}^{(d)}\leq u_{n}\}.$

By (21) in Markovich and Rodionov (2020) and assuming (5), we have

\displaystyle\sum_{i=d+1}^{l_{n}}P\{z_{i}M_{n}^{(i)}>u_{n}\}

\displaystyle=

\displaystyle o(1),~{}~{}n\to\infty.

(45)

It holds

			$\displaystyle P\{M_{n}^{}(z,d)\leq u_{n}\}=1-P\{M_{n}^{}(z,d)>u_{n}\}=1-P\{z_{d}M_{n}^{(d)}>u_{n}\}$
		$\displaystyle-$	$\displaystyle\sum_{j=1}^{d-1}P\{z_{j}M_{n}^{(j)}>u_{n},\max_{j+1\leq i\leq d}(z_{i}M_{n}^{(i)})\leq u_{n}\}.$

Note that $\sum_{j=1}^{d-1}P\{z_{j}M_{n}^{(j)}>u_{n},\max_{j+1\leq i\leq d}(z_{i}M_{n}^{(i)})\leq u_{n}\}=o(1)$ is equivalent to (17) for $i=d$ . If (17) holds, then we get

\displaystyle P\{M_{n}^{*}(z,d)\leq u_{n}\}=P\{z_{d}M_{n}^{(d)}\leq u_{n}\}+o(1),~{}~{}n\to\infty.

Then by (44) and (45) the expression

	$\displaystyle P\{M_{n}^{*}(z)\leq u_{n}\}$	$\displaystyle=$	$\displaystyle P\{z_{d}M_{n}^{(d)}\leq u_{n}\}(1+o(1))$
		$\displaystyle=$	$\displaystyle\exp(-\theta_{d}(z_{d}/y)^{k_{1}})(1+o(1))$

that is required to obtain the extremal index $\theta_{d}$ for $Y_{n}^{*}(z)$ follows by (3) and (29). For some $i\in\{1,...,d\}$ such that (17) holds, $Y_{n}^{*}(z)$ has the extremal index $\theta_{i}$ for reasons of symmetry. The same result holds for $Y_{n}(z)$ due to (32), (42) and (43) if the conditions (A2) are additionally fulfilled.

3.2 Proof of Theorem 2.2

3.2.1 Case (i)

Tail index

Similarly to the proof of Theorem 4 in Markovich and Rodionov (2020) we have for all $m\geq 1$

			$\displaystyle(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))\leq P\{z_{1}Y_{n,1}>x\}\leq P\{Y_{m}^{*}(z,N_{m})>x\}$		(46)
		$\displaystyle\leq$	$\displaystyle P\{Y_{m}(z,N_{m})>x\}\leq P\{Y_{m}(z,l(x))>x\}+P\{N_{m}>l(x)\},$		(46)

where $l(x)$ is selected in such a way to neglect the term $P\{N_{m}>l(x)\}$ taking into account (6). Namely, $l_{m}$ is replaced by an arbitrary natural number $l=l(x)$ and for all sufficiently large $x$ we set $l(x)=[x^{k_{1}/\alpha+\delta_{1}}]$ with $\delta_{1}>0$ .
We get

\displaystyle P\{Y_{m}^{*}(z,l(x))>x\}=P\{z_{1}Y_{n,1}>x\}+\sum_{i=2}^{l}P\{z_{i}Y_{n,i}>x,\max_{1\leq j\leq i-1}z_{j}Y_{n,j}\leq x\}.

Condition (A3)

Using notation (38) we obtain

\displaystyle\sum_{i=2}^{l}\xi_{i}

\displaystyle=

\displaystyle\sum_{s=2}^{\lfloor d_{n}-1\rfloor}\left(\sum_{i=2}^{s}\xi_{i}+\sum_{i=s+1}^{l}\xi_{i}\right)P\{d=s\}

(48)

since $d$ and $\{Y_{n,i}\}$ are assumed to be independent and $1<d<d_{n}=\min(C,l_{n})$ holds. By (33) we have

\displaystyle\sum_{i=s+1}^{l}\xi_{i}\leq\sum_{i=s+1}^{l}P\{z_{i}Y_{n,i}>x\}\leq O\left(x^{-k_{1}(1+\delta)}\right),~{}x\to\infty,

(49)

with $\delta\in(0,k_{1}\chi_{0})$ . By (A3) it holds

	$\displaystyle\sum_{i=2}^{s}\xi_{i}$	$\displaystyle=$	$\displaystyle\sum_{i=2}^{s}P\{z_{i}Y_{n,i}>x,\max_{1\leq j\leq i-1}z_{j}Y_{n,j}\leq x\}$		(50)
		$\displaystyle=$	$\displaystyle P\{\max_{2\leq j\leq s}(z_{j}Y_{n,j})>x,z_{1}Y_{n,1}\leq x\}=o(P\{Y_{n,1}>x\})$		(50)

for any $s\in\{2,3,...,\lfloor d_{n}-1\rfloor\}$ as $x\to\infty$ .
By (3.2.1)-(50) it follows

\displaystyle P\{Y^{*}_{m}(z,l(x))>x\}=P\{z_{1}Y_{n,1}>x\}(1+o(1))=(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1)).

(51)

Then by (46) $P\{Y_{m}^{*}(z,N_{m})>x\}$ has the tail index $k_{1}$ .

Condition (A1) (or (A2)) and the independence of $N_{m}$ and $\{Y_{m,i}\}$

If, in addition to condition (A1) or (A2), $N_{m}$ and $\{Y_{m,i}\}$ are assumed to be independent, then $P\{Y_{m}(z,N_{m})>x\}$ and $P\{Y^{*}_{m}(z,N_{m})>x\}$ have the same tail index $k_{1}$ . Really, by (33), (35), (36) (or (37)) we get for $m\geq 1$

			$\displaystyle P\{Y_{m}(z,N_{m})>x\}$		(52)
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{\infty}P\{Y_{m}(z,i)>x\}P\{N_{m}=i\}=x^{-k_{1}}\Omega(d_{m},k_{1})(1+o(1)),$		(52)

where

	$\displaystyle\Omega(d_{m},k_{1})=$
	$\displaystyle\!\!\!\!\!\sum_{i=1}^{\infty}\sum_{s=2}^{\lfloor d_{m}-1\rfloor}\left(\sum_{j=1}^{s}z_{j}^{k_{1}}\ell_{j}(x)\mathbf{1}\{i>s\}+\sum_{j=1}^{i}z_{j}^{k_{1}}\ell_{j}(x)\mathbf{1}\{i\leq s\}\right)P\{d=s\}P\{N_{m}=i\}.$

The same is valid for $P\{Y^{*}_{m}(z,N_{m})>x\}$ by the same reasons.

Extremal index

Let us find the extremal index of $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ . We use formula (31) in Markovich and Rodionov (2020)

$\displaystyle P\{z_{1}M_{n}^{(1)}>u_{n}\}$	$\displaystyle\leq$	$\displaystyle P\{M_{n}^{*}(z,N_{n})>u_{n}\}$	(53)
	$\displaystyle\leq$	$\displaystyle P\{M_{n}^{*}(z,\tilde{l}_{n})>u_{n}\}+P\{N>\tilde{l}_{n}\}$
	$\displaystyle\leq$	$\displaystyle P\{M_{n}^{*}(z,\tilde{l}_{n})>u_{n}\}+\sum_{i=1}^{n}P\{N_{i}>\tilde{l}_{n}\},$

where $\tilde{l}_{n}>l_{n}$ is selected such that the term $\sum_{i=1}^{n}P\{N_{i}>\tilde{l}_{n}\}\leq O(n^{1-\alpha(\chi+\delta)})=o(1)$ is neglected as $n\to\infty$ , i.e.

\displaystyle\tilde{l}_{n}=n^{\chi+2\delta},\qquad\delta=(\chi_{0}-\chi)/3.

(54)

Similarly to (3.2.1) we get

	$\displaystyle P\{M_{n}^{*}(z,\tilde{l}_{n})>u_{n}\}$	$\displaystyle=$	$\displaystyle P\{z_{1}M_{n}^{(1)}>u_{n}\}$
		$\displaystyle+$	$\displaystyle\sum_{i=2}^{\tilde{l}_{n}}P\{z_{i}M_{n}^{(i)}>u_{n},\max_{1\leq j\leq i-1}z_{j}M_{n}^{(j)}\leq u_{n}\}.$

Denoting $P\{z_{i}M_{n}^{(i)}>u_{n},\max_{1\leq j\leq i-1}z_{j}M_{n}^{(j)}\leq u_{n}\}$ as $\zeta_{i}$ and splitting the sum $\sum_{i=2}^{\tilde{l}_{n}}$ as in (48), we obtain

\displaystyle\sum_{s=2}^{\lfloor d_{n}-1\rfloor}\sum_{i=2}^{s}\zeta_{i}P\{d=s\}=o(1)

by (17) and

\displaystyle\sum_{s=2}^{\lfloor d_{n}-1\rfloor}\sum_{i=s+1}^{\tilde{l}_{n}}\zeta_{i}P\{d=s\}=o(1)

by (45) as $n\to\infty$ . (45) is valid for $\tilde{l}_{n}$ since $\chi+2\delta<\chi_{0}$ . Hence, it follows

\displaystyle P\{M_{n}^{*}(z,\tilde{l}_{n})>u_{n}\}=P\{z_{1}M_{n}^{(1)}>u_{n}\}(1+o(1)).

(55)

Then by (53)

\displaystyle P\{M_{n}^{*}(z,N_{n})>u_{n}\}=P\{z_{1}M_{n}^{(1)}>u_{n}\}(1+o(1))

follows and the extremal index of $Y_{n}^{*}(z,N_{n})$ is equal to $\theta_{1}$ (or for reasons of symmetry to $\theta_{i}$ for some $i\in\{1,...,d\}$ such that (17) holds). $Y_{n}(z,N_{n})$ has the same extremal index since $P\{M_{n}^{*}(z,N_{n})>u_{n}\}\leq P\{M_{n}(z,N_{n})>u_{n}\}\leq P\{M_{n}(z,\tilde{l}_{n})>u_{n}\}+\sum_{i=1}^{n}P\{N_{i}>\tilde{l}_{n}\}$ by (53) holds, and since, in addition, the condition (A1) or (A2) is assumed due to (32), (39), (41), (42) and (43).

3.2.2 Case (ii)

Let $d=d_{0}$ a.s. Then the same statements as in Case (i) follow. It is enough to replace $s$ in formulas of Case (i) by $d_{0}$ with probability one.

3.3 Proof of Corollary 1

3.3.1 Case $k_{1}^{+}\leq k_{1}^{-}$

We assume that there is a unique ”column” series with a minimum tail index $k_{1}^{+}$ . One can derive that

	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{+}\}=(z_{1}^{+}/y)^{k_{1}^{+}}n^{-1}(1+o(1)),$
	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{-}\}=(z_{1}^{+}/y)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}(1+o(1))$		(56)

as $n\to\infty$ . Really, since $Y_{n}(z,N_{n})\leq Y_{n}^{+}(z^{+},N_{n}^{+})$ holds, then by (23), (29), (31) with $d=1$ it holds

			$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{+}\}\leq P\{Y_{n}^{+}(z^{+},l_{n^{+}})>u_{n}^{+}\}+P\{N_{n}^{+}>l_{n^{+}}\}$		(57)
		$\displaystyle=$	$\displaystyle P\{Y_{n}^{+}(z^{+},l_{n^{+}})>u_{n}^{+}\}(1+o(1))=(z_{1}^{+}/y)^{k_{1}^{+}}n^{-1}(1+o(1)),$		(57)

	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{-}\}$	$\displaystyle\leq$	$\displaystyle P\{Y_{n}^{+}(z^{+},l_{n^{+}})>u_{n}^{-}\}(1+o(1))$		(58)
		$\displaystyle=$	$\displaystyle(z_{1}^{+}/y)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}(1+o(1)).$		(58)

Let us obtain the lower bounds of $P\{Y_{n}(z,N_{n})>u_{n}^{\pm}\}$ . Since it holds

			$\displaystyle P\{-Y_{n}^{-}(z^{-},N_{n}^{-})>u_{n}^{+}\varepsilon\}$
		$\displaystyle=$	$\displaystyle P\{-Y_{n}^{-}(z^{-},N_{n}^{-})>u_{n}^{+}\varepsilon,N_{n}^{-}\leq l_{n^{-}}\}+P\{-Y_{n}^{-}(z^{-},N_{n})>u_{n}^{+}\varepsilon,N_{n}^{-}>l_{n^{-}}\}$
		$\displaystyle\leq$	$\displaystyle P\{Y_{n}^{-}(\|z^{-}\|,l_{n^{-}})>u_{n}^{+}\varepsilon\}+P\{N_{n}^{-}>l_{n^{-}}\}$

and by (23), (29), (31) we get for arbitrary $\varepsilon>0$ , $d=1$ and $k_{1}^{+}\leq k_{1}^{-}$

	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{+}\}=P\{Y_{n}^{+}(z^{+},N_{n}^{+})+Y_{n}^{-}(z^{-},N_{n}^{-})>u_{n}^{+}\}$	(59)
$\displaystyle\geq$	$\displaystyle P\{Y_{n}^{+}(z^{+},N_{n}^{+})>u_{n}^{+}(1+\varepsilon),0<-Y_{n}^{-}(z^{-},N_{n}^{-})\leq u_{n}^{+}\varepsilon\}$
$\displaystyle\geq$	$\displaystyle P\{Y_{n}^{+}(z^{+},N_{n}^{+})>u_{n}^{+}(1+\varepsilon)\}-P\{-Y_{n}^{-}(z^{-},N_{n}^{-})>u_{n}^{+}\varepsilon\}$
$\displaystyle\geq$	$\displaystyle P\{z_{1}^{+}Y_{n,1}>u_{n}^{+}(1+\varepsilon)\}-P\{Y_{n}^{-}(\|z^{-}\|,l_{n^{-}})>u_{n}^{+}\varepsilon\}-P\{N_{n}^{-}>l_{n^{-}}\}$
$\displaystyle=$	$\displaystyle\left(\left(\frac{z_{1}^{+}}{y(1+\varepsilon)}\right)^{k_{1}^{+}}n^{-1}-\left(\frac{\|z_{1}^{-}\|}{y\varepsilon}\right)^{k_{1}^{-}}n^{-k_{1}^{-}/k_{1}^{+}}\right)(1+o(1))$
$\displaystyle=$	$\displaystyle\left(\frac{z_{1}^{+}}{y(1+\varepsilon)}\right)^{k_{1}^{+}}n^{-1}(1+o(1)),$

	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{-}\}$	$\displaystyle\geq$	$\displaystyle\left(\left(\frac{z_{1}^{+}}{y(1+\varepsilon)}\right)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}-\left(\frac{\|z_{1}^{-}\|}{y\varepsilon}\right)^{k_{1}^{-}}n^{-1}\right)(1+o(1))$		(60)
		$\displaystyle=$	$\displaystyle\left(\frac{z_{1}^{+}}{y(1+\varepsilon)}\right)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}(1+o(1)).$		(60)

In the same way, (3.3.1) is true for $Y^{*}_{n}(z,N_{n})$ .

Tail index

Let us show that the sequence $Y_{n}(z,N_{n})$ in (2.2) has the tail index $k_{1}^{+}$ for $k_{1}^{+}\leq k_{1}^{-}$ . As in the proof of Theorem 2.2 let us replace $l_{m^{\pm}}$ by an arbitrary natural number $l^{\pm}=l^{\pm}(x)=[x^{k_{1}^{\pm}/\alpha+\delta_{1}}]$ with $\delta_{1}>0$ for all sufficiently large $x$ . By Theorem 1.1 we have

	$\displaystyle P\{Y_{m}(z,N_{m})>x\}\leq P\{Y_{m}^{+}(z^{+},l^{+}(x))>x\}+P\{N_{m}^{+}>l^{+}(x)\}$	(61)
$\displaystyle=$	$\displaystyle(z_{1}^{+}/x)^{k_{1}^{+}}\ell_{1}(x)(1+o(1))+x^{-k_{1}^{+}-\alpha\delta_{1}}\widetilde{\ell}_{m}(x^{k_{1}^{+}/\alpha+\delta_{1}})$
$\displaystyle=$	$\displaystyle(z_{1}^{+}/x)^{k_{1}^{+}}\ell_{1}(x)(1+o(1))$

as $x\to\infty$ for all $m\geq 1$ . On the other hand, we get similarly to (59)

	$\displaystyle P\{Y_{m}(z,N_{m})>x\}$	(62)
$\displaystyle\geq$	$\displaystyle P\{z_{1}^{+}Y_{m,1}>x(1+\varepsilon)\}-P\{Y_{m}^{-}(\|z^{-}\|,l_{m^{-}})>x\varepsilon\}-P\{N_{m}^{-}>l_{m^{-}}\}$
$\displaystyle=$	$\displaystyle\left(\left(\frac{z_{1}^{+}}{x(1+\varepsilon)}\right)^{k_{1}^{+}}-\left(\frac{\|z_{1}^{-}\|}{x\varepsilon}\right)^{k_{1}^{-}}\right)\ell_{1}(x)(1+o(1))-x^{-k_{1}^{-}-\alpha\delta_{1}}\widetilde{\ell}_{m}(x^{k_{1}^{-}/\alpha+\delta_{1}})$
$\displaystyle=$	$\displaystyle(z_{1}^{+}/x)^{k_{1}^{+}}\ell_{1}(x)(1+o(1)).$

The same is valid for $Y^{*}_{m}(z,N_{m})$ . Then the first statement follows.

Extremal index

By (3.3.1) the extremal index of $Y_{n}(z,N_{n})$ and of $Y^{*}_{n}(z,N_{n})$ does not exist for $u_{n}^{-}$ due to (2).
By (2.2) and Theorem 1.2 it follows that $Y_{n}^{*}(z,N_{n})$ has the extremal index $\theta^{+}_{1}$ corresponding to $k_{1}^{+}$ for $u_{n}^{+}$ . The same holds for $Y_{n}(z,N_{n})$ since by (42) and (53)

\displaystyle P\{M_{n}^{*}(z,N_{n})\leq u_{n}\}

\displaystyle=

\displaystyle P\{M_{n}(z,N_{n})\leq u_{n}\}(1+o(1))

(63)

follows. Really, similarly to (42) we get

$\displaystyle 0$	$\displaystyle\leq$	$\displaystyle P\{M_{n}^{*}(z,l_{n})\leq u_{n}\}-P\{M_{n}(z,l_{n})\leq u_{n}\}$
	$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}P\{Y_{j}^{*+}(z^{+})\leq u_{n},Y_{j}^{-}(z^{-})+Y_{j}^{+}(z^{+})>u_{n}\}$
	$\displaystyle\leq$	$\displaystyle\sum_{j=1}^{n}P\{Y_{j}^{*+}(z^{+})\leq u_{n},Y_{j}^{+}(z^{+})>u_{n}\},$

where $Y_{n}(z)$ and $Y_{n}^{*}(z)$ denote $Y_{n}(z,l_{n})$ and $Y_{n}^{*}(z,l_{n})$ . We obtain

			$\displaystyle P\{Y_{j}^{+}(z^{+})\leq u_{n},Y_{j}^{+}(z^{+})>u_{n}\}=P\{Y_{j}^{+}(z^{+})>u_{n}\}-P\{Y_{j}^{+}(z^{+})>u_{n}\}$		(64)
		$\displaystyle\leq$	$\displaystyle P\{Y_{n}^{+}(z^{+})>u_{n}\}-P\{z_{1}^{+}Y_{n,1}>u_{n}\}=o(1/n)$		(64)

by (29) and (31) with $d=1$ . Then $P\{M_{n}^{*}(z,l_{n})\leq u_{n}^{+}\}=P\{M_{n}(z,l_{n})\leq u_{n}^{+}\}(1+o(1))$ holds and by (22) in Markovich and Rodionov (2020) $P\{M_{n}^{*}(z,l_{n})\leq u_{n}^{+}\}=P\{z_{1}M_{n}^{(1)}\leq u_{n}^{+}\}(1+o(1))$ holds. Then (63) follows by (53).

3.3.2 Case $k_{1}^{-}\leq k_{1}^{+}$

We assume that there is a unique ”column” series with the minimum tail index $k_{1}^{-}$ . Similarly to (3.3.1) one can get

	$\displaystyle P\{-Y_{n}(z,N_{n})>u_{n}^{-}\}=(\|z_{1}^{-}\|/y)^{k_{1}^{-}}n^{-1}(1+o(1)),$
	$\displaystyle P\{-Y_{n}(z,N_{n})>u_{n}^{+}\}=(\|z_{1}^{-}\|/y)^{k_{1}^{-}}n^{-k_{1}^{-}/k_{1}^{+}}(1+o(1)),~{}~{}n\to\infty,$		(65)

and by (61), (62) for all $m\geq 1$

\displaystyle P\{-Y_{m}(z,N_{m})>x\}=(|z_{1}^{-}|/x)^{k_{1}^{-}}\ell_{1}(x)(1+o(1))

as $x\to\infty$ . The same is valid for $-Y_{n}^{**}(z,N_{n})$ . Then $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ in (2.2) have the same tail index $k_{1}^{-}$ .
By (3.3.2) the extremal indices of $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ do not exist for $u_{n}^{+}$ due to (2). In the same way as for the case $k_{1}^{+}\leq k_{1}^{-}$ one can derive that $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ have the same extremal index equal to $\theta^{-}_{1}$ for $u_{n}=u_{n}^{-}$ .

3.4 Proof of Corollary 2

3.4.1 Case $k_{1}^{+}\leq k_{1}^{-}$

By (2.2) all statements of Theorem 2.2 for $Y_{n}^{*}(z,N_{n})$ regarding the tail index and the extremal index in case of $u_{n}=u_{n}^{+}$ are fulfilled. It remains to show the same for $Y_{n}(z,N_{n})$ .

Tail index

One can derive that the tail index of $Y_{n}(z,N_{n})$ is given by $k_{1}^{+}$ in the same way as in Theorem 2.2 assuming that $N_{m}^{\pm}$ and $\{Y^{\pm}_{m,i}\}$ are independent and (A1) (or (A2)) holds. Really, by (52) and similarly to (59) we get

\displaystyle P\{Y_{m}(z,N_{m})>x\}\leq P\{Y^{+}_{m}(z^{+},N^{+}_{m})>x\}=x^{-k^{+}_{1}}\ell_{1}(x)\Omega(d^{+}_{m},k^{+}_{1})(1+o(1)),

$\displaystyle P\{Y_{m}(z,N_{m})>x\}$	$\displaystyle\geq$	$\displaystyle P\{Y_{m}^{+}(z^{+},N_{m}^{+})>x(1+\varepsilon)\}-P\{-Y_{m}^{-}(z^{-},N_{m}^{-})>x\varepsilon\}$
	$\displaystyle=$	$\displaystyle P\{Y_{m}^{+}(z^{+},N_{m}^{+})>x(1+\varepsilon)\}-P\{Y_{m}^{-}(\|z^{-}\|,N_{m}^{-})>x\varepsilon\}$
	$\displaystyle=$	$\displaystyle(x^{-k^{+}_{1}}\ell_{1}(x)\Omega(d^{+}_{m},k^{+}_{1})-x^{-k^{-}_{1}}\ell_{1}(x)\Omega(d^{-}_{m},k^{-}_{1}))(1+o(1))$
	$\displaystyle=$	$\displaystyle x^{-k^{+}_{1}}\ell_{1}(x)\Omega(d^{+}_{m},k^{+}_{1})(1+o(1))$

for any $m\geq 1$ .

Extremal index

Let us find the extremal index of $Y_{n}(z,N_{n})$ . Since $Y_{n}(z,N_{n})\leq Y_{n}^{+}(z^{+},N_{n}^{+})$ holds, then it follows by (23), (29), (31) and (58)

\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{-}\}\leq\left(\frac{z^{*+}}{y(1-\varepsilon)}\right)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}(1+o(1))+o(1/n),

(66)

where $(z^{*+})^{k_{1}^{+}}=\sum_{i=1}^{d^{+}}(z_{i}^{+}/\varepsilon_{i}^{*})^{k_{1}^{+}}$ , $u_{n}^{\pm}$ are determined as in Corollary 1. By (60) we obtain

\displaystyle P\{Y_{n}(z,N_{n})>u_{n}^{-}\}\geq\left(\frac{z_{1}^{+}}{y}\right)^{k_{1}^{+}}n^{-k_{1}^{+}/k_{1}^{-}}(1+o(1)).

(67)

The extremal index of $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ does not exist for $u_{n}^{-}$ due to (2), (66) and (67). $Y_{n}(z,N_{n})$ has the same extremal index as $Y^{*}_{n}(z,N_{n})$ for $u_{n}^{+}$ since (63) holds. Really, note that

\displaystyle P\{M_{n}^{*}(z,l_{n})\leq u_{n}^{+}\}=P\{M_{n}(z,l_{n})\leq u_{n}^{+}\}(1+o(1))

(68)

holds since (64) is fulfilled by (A1) (see, (39), (41)) or (A2) (see, (32)). Then (63) follows by (53) and (55) in the same way as in Theorem 2.2.

3.4.2 Case $k_{1}^{-}\leq k_{1}^{+}$

The proof of the tail index for $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ is similar to the previous case. The extremal index of $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ does not exist for $u_{n}^{+}$ . The extremal index for $-Y_{n}(z,N_{n})$ and $-Y_{n}^{**}(z,N_{n})$ and $u_{n}=u_{n}^{-}$ is determined symmetrically as for the case $k_{1}^{+}\leq k_{1}^{-}$ .

3.5 Proof of Theorem 2.3

The proof is similar to that of Theorem 1.2. We only indicate the modifications.

Case (i)

In conditions of Theorem 1.2 it holds

	$\displaystyle P\{Y_{n}(z,l_{n})>u_{n}\}$	$\displaystyle=$	$\displaystyle P\{Y_{n}^{*}(z,l_{n})>u_{n}\}(1+o(1))=P\{z_{1}Y_{n,1}>u_{n}\}(1+o(1))$		(69)
		$\displaystyle=$	$\displaystyle(z_{1}/y)^{k_{1}}n^{-1}(1+o(1))$		(69)

as $\ n\to\infty$ (see also (8) and (10) in Markovich and Rodionov (2020)). Hence, (11) implies

\displaystyle P\{Y_{n}(z,l_{n})>u_{n}\}

\displaystyle=

\displaystyle o\left(P\{N_{n}>l_{n}\}\right).

Indeed, $\alpha\chi<1$ follows by (11).

Tail index

Let us find the tail index of $Y^{*}_{n}(z,N_{n})$ and $Y_{n}(z,N_{n})$ . The claim $\alpha\chi_{0}>1$ is required in Theorem 1.2 for the following result derived in formula (15) in Markovich and Rodionov (2020)

			$\displaystyle P\{Y_{m}(z,l_{m})>x\}=P\{Y_{m}^{*}(z,l_{m})>x\}(1+o(1))$
		$\displaystyle=$	$\displaystyle P\{Y_{m,1}>x/z_{1}\}(1+o(1))=(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))$

for any fixed $m\geq 1$ and as $x\to\infty$ . To this end, we use $l(x)=[x^{k_{1}/\alpha+\delta_{1}}]$ with $\delta_{1}>0$ to replace $l_{m}$ in (16) by Markovich and Rodionov (2020) for sufficiently large $x$ such that $l(x)<x^{k_{1}\chi_{0}-\delta_{2}}$ with arbitrary $\delta_{2}\in(0,k_{1}\chi_{0})$ , where $\delta_{1}+\delta_{2}=\delta^{*}$ , $\delta^{*}$ is determined by (24). Thus, by (24) $x^{k_{1}/\alpha+\delta_{1}}<x^{k_{1}\chi_{0}-\delta_{2}}$ holds for $x>1$ . In the same way as in (34) for $d=1$ and by (1) we have

$\displaystyle P\{Y_{m}^{*}(z,N_{m})>x\}$	$\displaystyle\leq$	$\displaystyle P\{Y_{m}(z,N_{m})>x\}$
	$\displaystyle\leq$	$\displaystyle P\{Y_{m}(z,l(x))>x\}+P\{N_{m}>l(x)\}$
	$\displaystyle=$	$\displaystyle(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))+x^{-k_{1}-\alpha\delta_{1}}\tilde{\ell}_{m}(x^{k_{1}/\alpha+\delta_{1}})$
	$\displaystyle=$	$\displaystyle(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1)),$

	$\displaystyle P\{Y_{m}^{*}(z,N_{m})>x\}$	$\displaystyle\geq$	$\displaystyle P\{Y_{m,1}>x/z_{1}\}(1+o(1))$		(71)
		$\displaystyle=$	$\displaystyle(z_{1}/x)^{k_{1}}\ell_{1}(x)(1+o(1))$		(71)

as $x\to\infty$ . Then the first statement of (i) in Theorem 2.3 follows.

Extremal index

Let us find the extremal index of $Y_{m}(z,N_{m})$ and $Y_{m}^{*}(z,N_{m})$ . Let us use $\widetilde{l}_{n}$ as in (54). By (7) $\widetilde{l}_{n}>l_{n}$ follows and (69) holds for $\widetilde{l}_{n}$ since $\chi+2\delta<\chi_{0}$ holds. By (6), (7), (11), (29), (69) we have

			$\displaystyle(z_{1}/y)^{k_{1}}n^{-1}(1+o(1))=P\{z_{1}Y_{n,1}>u_{n}\}\leq P\{Y^{*}_{n}(z,N_{n})>u_{n}\}$
		$\displaystyle\leq$	$\displaystyle P\{Y_{n}(z,N_{n})>u_{n}\}\leq P\{Y_{n}(z,\widetilde{l}_{n})>u_{n}\}+P\{N_{n}>\widetilde{l}_{n}\}$
		$\displaystyle\leq$	$\displaystyle(z_{1}/y)^{k_{1}}n^{-1}(1+o(1))+n^{-\alpha(\chi+2\delta)}\tilde{\ell}_{n}(n^{\chi+2\delta})$
		$\displaystyle=$	$\displaystyle(z_{1}/y)^{k_{1}}n^{-1}(1+o(1)),~{}~{}n\to\infty$

since

\displaystyle\alpha(\chi+2\delta)>1

(72)

is valid by (25) and $\alpha\chi<1$ .
Let us denote $\tilde{l}_{n}^{*}=n^{\chi+2.5\delta}$ . By (31) in Markovich and Rodionov (2020) we get

	$\displaystyle P\{z_{1}M_{n}^{(1)}>u_{n}\}$	$\displaystyle\leq$	$\displaystyle P\{M_{n}^{}(z,N_{n})>u_{n}\}\leq P\{M_{n}^{}(z,\tilde{l}_{n}^{})>u_{n}\}+P\{N>\tilde{l}_{n}^{}\}$		(73)
		$\displaystyle\leq$	$\displaystyle P\{M_{n}^{}(z,\tilde{l}_{n}^{})>u_{n}\}+\sum_{i=1}^{n}P\{N_{i}>\tilde{l}_{n}^{*}\}.$		(73)

We get

	$\displaystyle\sum_{i=1}^{n}P\{N_{i}>\tilde{l}_{n}^{*}\}$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}n^{-\alpha(\chi+2\delta)}n^{-0.5\alpha\delta}\tilde{\ell}_{i}(n^{\chi+2.5\delta})$		(74)
		$\displaystyle\leq$	$\displaystyle O(n^{1-\alpha(\chi+2\delta)})=o(1)$		(74)

since $n^{-0.5\alpha\delta}\tilde{\ell}_{i}(n^{\chi+2.5\delta})\to 0$ uniformly for all $i\in\{1,...,n\}$ as $n\to\infty$ and (72) holds. By (22) in Markovich and Rodionov (2020), i.e. $P\{M_{n}^{*}(z)\leq u_{n}\}=P\{z_{1}M_{n}^{(1)}\leq u_{n}\}(1+o(1))$ , (73) and (74) it follows

\displaystyle\lim_{n\to\infty}P\{M_{n}^{*}(z,N_{n})\leq u_{n}\}=\exp\{-\theta_{1}(z_{1}/y)^{k_{1}}\}

and the extremal index of $Y^{*}_{n}(z,N_{n})$ is equal to $\theta_{1}$ by (69). It remains to show the same for $Y_{n}(z,N_{n})$ . To this end, one can derive

\displaystyle\lim_{n\to\infty}P\{M_{n}(z,N_{n})\leq u_{n}\}

\displaystyle=

\displaystyle\lim_{n\to\infty}P\{M_{n}^{*}(z,N_{n})\leq u_{n}\}

by (73), (74) and since $P\{M_{n}^{*}(z)\leq u_{n}\}=P\{M_{n}(z)\leq u_{n}\}(1+o(1))$ holds by (23) in Markovich and Rodionov (2020).

Case (ii)

The tail index of $P\{Y_{m}(z,N_{m})>x\}$ and $P\{Y_{m}^{*}(z,N_{m})>x\}$ is equal to $k_{1}$ by (3.5) and (71). Let us find their extremal index. By (26) $\alpha\chi=1$ and then (74) hold. Thus, the statements of Item (i) regarding the extremal index are fulfilled.

3.6 Proof of Theorem 2.4

Tail index

If (A3) holds, then $Y_{n}^{*}(z,N_{n})$ has the tail index $k_{1}$ by (51) and (3.5). As in the proof of Theorem 2.2 the sequences $Y_{n}^{*}(z,N_{n})$ and $Y_{n}(z,N_{n})$ have the same tail index $k_{1}$ if $N_{n}$ and $\{Y_{n,i}\}$ are independent and (A1) (or (A2)) holds by (52).

Extremal index

By (55), (72), (73) and (74) it follows $P\{M^{*}_{n}(z,N_{n})>u_{n}\}=P\{z_{1}M^{(1)}_{n}>u_{n}\}(1+o(1))$ . The same is valid for $P\{M_{n}(z,N_{n})$ by the same reasoning as in Theorem 2.2.

Acknowledgements.

The author was supported by the Russian Science Foundation (grant No. 22-21-00177). The author would like to thank Igor Rodionov for his useful comments.

References

(1) Asmussen, S., Foss, S.: Regular variation in a fixed-point problem for single- and multi-class branching processes and queues. Branching Processes and Applied Probability. Papers in Honour of Peter Jagers. Adv. Appl. Prob. 50A, 47-61 (2018)
(2) Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular variation, Cambridge university press, Cambridge (1987)
(3) Davis, R. A., Resnick, S. I.: Limit Theory for Bilinear Processes with Heavy-Tailed Noise. The Annals of Applied Probability, 6(4), 1191-1210 (1996)
(4) Jessen, A. H., Mikosch, T.: Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80, 171-192 (2006)
(5) Goldaeva, A. A.: Indices of multivariate recurrent stochastic sequences. In book ed. Shiryaev A. N.: Modern problem of mathematics and mechanics VIII(3), 42-51, Moscow State University ISBN 978–5–211–05652–7 (2013) (in Russian)
(6) Goldaeva, A. A., Lebedev, A. V.: On extremal indices greater than one for a scheme of series. Lith. Math. J. 58, 384-398 (2018)
(7) Leadbetter, M.R., Lingren, G., Rootz $\acute{e}$ n, H.: Extremes and Related Properties of Random Sequence and Processes. ch.3, Springer, New York (1983)
(8) Lebedev, A. V.: Activity maxima in some models of information networks with random weights and heavy tails. Problems of Information Transmission 51(1), 66-74 (2015a)
(9) Lebedev, A. V.: Extremal indices in the series scheme and their applications. Informatics Appl. 9(3), 39-54. (in Russian) (2015b)
(10) Markovich, N.M., Rodionov I.V. Maxima and sums of non-stationary random length sequences. Extremes, 23(3), 451-464 DOI: 10.1007/s10687-020-00372-5 (2020)
(11) Olvera-Cravioto, M.: Asymptotics for weighted random sums. Adv. Appl. Prob. 44(4), 1142-1172 (2012)
(12) Robert, C.Y., Segers, J.: Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance: Mathematics and Economics. 43, 85-92 (2008)
(13) Tillier, C., Wintenberger, O.: Regular variation of a random length sequence of random variables and application to risk assessment. Extremes. 21, 27-56 (2018)

WEIGHTED MAXIMA AND SUMS OF NON-STATIONARY RANDOM LENGTH SEQUENCES IN HEAVY-TAILED MODELS

Abstract

Keywords:

pacs:

1 Introduction

Definition 1

Theorem 1.1

Theorem 1.2

Remark 1

2 Main Results

2.1 Revision of Theorem 1.2 for positive constant weights

Example 1

Example 2

Theorem 2.1

Remark 2

Theorem 2.2

2.2 Revision of Theorem 1.2 for real-valued constant weights

2.2.1 A unique ”column” series with a minimum tail index

Corollary 1

2.2.2 A random number of ”column” series with a minimum tail index

Corollary 2

2.3 Revision of Theorem 1.2 for a heavy-tailed number of light-tailed terms

2.3.1 A unique ”column” series with a minimum tail index

Theorem 2.3

2.3.2 A random number of ”column” series with a minimum tail index

Theorem 2.4

3 Proofs

3.1 Proof of Theorem 2.1

Tail index

Condition (A1)

Conditions (A2)

Condition (A3)

The extremal index for dd independent ”column” sequences

The extremal index for dd dependent ”column” sequences

3.2 Proof of Theorem 2.2

3.2.1 Case (i)

Tail index

Condition (A3)

Condition (A1) (or (A2)) and the independence of NmN_{m} and {Ym,i}\{Y_{m,i}\}

Extremal index

3.2.2 Case (ii)

3.3 Proof of Corollary 1

3.3.1 Case k1+≤k1−k_{1}^{+}\leq k_{1}^{-}

Tail index

Extremal index

3.3.2 Case k1−≤k1+k_{1}^{-}\leq k_{1}^{+}

3.4 Proof of Corollary 2

3.4.1 Case k1+≤k1−k_{1}^{+}\leq k_{1}^{-}

Tail index

Extremal index

3.4.2 Case k1−≤k1+k_{1}^{-}\leq k_{1}^{+}

3.5 Proof of Theorem 2.3

Case (i)

Tail index

Extremal index

Case (ii)

3.6 Proof of Theorem 2.4

Tail index

Extremal index

Acknowledgements.

References

The extremal index for $d$ independent ”column” sequences

The extremal index for $d$ dependent ”column” sequences

Condition (A1) (or (A2)) and the independence of $N_{m}$ and $\{Y_{m,i}\}$

3.3.1 Case $k_{1}^{+}\leq k_{1}^{-}$

3.3.2 Case $k_{1}^{-}\leq k_{1}^{+}$

3.4.1 Case $k_{1}^{+}\leq k_{1}^{-}$

3.4.2 Case $k_{1}^{-}\leq k_{1}^{+}$