Introduction

Let and let be a collection of possibly dependent real-valued random variables (r.v.s) with heavy-tailed distributions. Denote

Throughout the paper, we assume that random summands have consistently varying distributions. This is a subclass of heavy-tailed distributions. We recall some definitions. We say that a distribution function (d.f.) is supported on if its tail satisfies

• A d.f. F supported on R is said to be heavy-tailed, written as F , if for every it holds that

• A d.f. F supported on is said to be heavy-tailed, written as , if for every it holds that

• A d.f. F on R is said to be dominatedly vaying as , if for any fixed (0,1)it holds that

• A d.f. F on is said to be consistently varying, written as, if

• A d.f. F on is said to be regularly varying with index , written as if for any, it holds that

It is well known (see, for instance, [5]) that

The following two indices are important to the determination whether d.f. F belongs to the aforementioned heavy-tailed distribution classes. The first index is the so-called upper Matuszewska index (see, e.g., [2, Sect. 2], [9, 23]), defined as

Another index, so-called L-index, is defined as

This index was used by [16, 19, 33], among others.

The definitions of the aforementioned heavy-tailed distribution classes imply that

The classes and have been extensively used in real analysis and various areas of probability (see, e.g., [2, 12, 25, 27]). The class of consistently varying distributions was introduced as a generalization of the class in [8], and was named there as a class of distributions with “intermediate regular variation”. The concept of consistent variation has been used in various papers in the context of applied probability, such as queueing systems, graph theory and ruin theory (see, e.g., [1, 3–7, 9, 13, 17, 22, 32]).

We explain some notations which will be used throughout the paper. For two positive functions f , g, we write:

In this paper, we suppose that the random variables are pairwise quasi- asymptotically independent. This dependence structure was introduced in [7] and consid- ered in [14, 20, 21, 30, 31] and other papers. In the definition below and elsewhere, we use the standard notations:

Definition 1. Real-valued random variables with distributions supported on are called pairwise quasi-asymptotically independent (pQAI) if for all pairs of indices it holds that

The following statement is Theorem 3.1 in [7]. The statement provides the asymptotic results for tail probability of sums of r.v.s having distributions from class

Theorem 1.Let be a collection of real-valued r.v.s such that Then

The following assertion with slightly narrower dependence structure and r.v.s from a wider class is derived in Theorem 2.1 of [18].

Theorem 2. Let be a collection of real-valued r.v.s such that

for all pairs of indices In addition, suppose that for some

Then

In this paper, we obtain asymptotic relationships for

and

for arbitrary power α [0, ) and for r.v.s following wider, , depen- dence structure. Asymptotic behavior of the left truncated moments of random sums was considered in various fields of applied probability, including risk theory and random walks [10,11,24]. In addition, quantity in (3) is closely related with the Haezendonck–Goovaerts risk measure (see, for instance, [15, 18, 28] and [29]). To get the precise asymptotic equivalence relationship, we consider r.v.s with d.f.s from class The main results on the asymptotics of (2) and (3) are presented in Theorems 3 and 4 below.

The rest of the paper is organized as follows. In Section 2, we provide formulations of the main results. In Section 3, we present the proofs of the asymptotic formulas for the left truncated moments of . The last Section 4 deals with the examples illustrating the obtained results.

Main results

The first assertion generalizes results of Theorem 1 which can be derived from theorem below by supposing . In addition, for class, theorem below gives an analogous result to Theorem 2 for r.v.sfollowing a wider dependence structure and for a real-valued nonnegative moment order α.

Theorem 3. Let be a collection of real-valued r.v.s such that and Then

The second theorem shows that the asymptotic behaviour of the left truncated mo- ments of sums depends on consistently varying distributed increments but does not depend on asymptotically lighter increments.

There 4. let be a collectin of real-valued r,v.s such that ,for each , it holds that . Suppose that and and some be a subset of indices k such that . If the subcollection consists of r.v.s, then,

for each

and, for it holds that

We notice that the basic index in the formulation of Theorem 4, which is equal to one, can be replaced by any index . In addition, it should be noted that depen- dence of r.v.s, as well as mutual dependence between the sets and , can be arbitrary.

Proofs of main results

We present two auxiliary lemmas before providing proofs of the main results.

Lemma 1. Let be a real-valued r.v. such that for some Then, for any , we have

and

Proof. Both equalities of the lemma follow directly from the following well-known formula

provided that and η is a nonnegative r.v. (see, for instance, [26, p. 208, Cor. 2]).

Namely, by supposing, from (9) we obtain

and equality (7) follows.

Similarly, by supposing , from (9) equality (8) holds because

Lemma 2. Let and η be two arbitrarily dependent r.v.s. If and then

Proof. Proof of the lemma is presented in [34] (see part (i) of Lemma 3.3).

Proof of Theorem 3. In the case the assertion of Theorem 3 follows from Theo- rem 1 immediately. Hence, further, we can suppose that α is positive. By Lemma 1, for all , we have

due to right inequality in min-max inequality

provided that and .

By Theorem 1 we get

Similarly, using the left inequality in (11), we obtain

The derived estimates (12) and (13) complete the proof of Theorem 3.

Proof of Theorem 4. If , then relation (5) follows immediately from The- orem 3. Hence, let us suppose that and denote

Summands in are r.v.s with consistently varying d.f.s. Hence, Theorem 1 implies that

This asymptotic relation and inequality (11) imply that d.f. belongs to the class due to the following estimate

provided that y (0, 1).

In addition, each r.v. _k with index satisfies condition according to requirements of the theorem. The fact that Fξ1 and asymptotic equality (14) imply that

because

where

Consequently, Lemma 2 and asymptotic relations (14), (15) imply that

Hence, the first relation (5) of Theorem 4 holds in the case then using the first equality of Lemma 1 and estimates of (11), similarly as in the proof of

Theorem 3, we derive that

Relation (5) of Theorem 4 forfollows now from (16).

Importar imagenThe second asymptotic relation (6) can be obtained in a similar way by using the second equality of Lemma 1, relation (16) and estimate (11). Theorem 4 is proved.

Examples

In this section, we provide two examples illustrating our main results.

Example 1. Let r.v.s satisfy the assumptions of Theorem 3. Suppose that for each k, r.v. is a copy of r.v., where are independent, is uniformly distributed on interval [0, 1], and is geometrically distributed with parameter q ∈ (0, 1), i.e.,. We derive the asymptotic formulas for

in the case of 0 ≤ α < log2(1/q), where as usual.

Due to considerations on pages 122–123 of [5], . In addition, for , we have

where symbol denotes the integer part of a real number a, symbol denotes the fractional part of a, and function f is defined by the following equality

For the function f , we have

Consequently, for ,

where B denotes the Beta function

These relations and theorems 3, 4 imply that

for (0, 1) and and

for all (0, 1) and

The derived asymptotic formulas imply the following particular cases:

if q ∈ (0, 1/4).

Example 2. Let r.v.s , be pQAI. Suppose that is distributed according to the following tail function

For other indices let us suppose that

Like in Example 1, we write asymptotic formulas for the left truncated moments

in the case of suitable α.

It is obvious that , and, further,du to results of [9] (see page 87)

Therefore, Theorem 4 implies that

and

Consequently,

and, for α ∈ (0, 1),

Acknowledgments

We would like to thank the four anonymous referees for the detailed and helpful comments on the previous versions of the manuscript.

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Notes