Regularity of a Randomly Stopped Sum Determines Regularity of the Stopping Moment

This paper is devoted to randomly stopped sums in which the primary random variables (r.v.s) are nonnegative, independent, and identically distributed (i.i.d.). The stopping moment is supposed to be nonnegative, integer-valued, and independent of the primary r.v.s. Such objects appear when the number of random variables under consideration is unknown and is described by some random integer. In particular, randomly stopped sums appear in insurance and financial mathematics, survival analysis, risk theory, computer and communication networks, etc. The area of randomly stopped sums for various subclasses of heavy-tailed r.v.s has been well developed for more than 50 years and covers mainly the case of i.i.d. r.v.s. In this paper, we continue to consider this standard model.

Specifically, suppose that $\{{\xi }_1,{\xi }_2,\cdots \}$ is a sequence of r.v.s defined on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$. Define the sequence of partial sums $\{S_n,n⩾0\}$ by

The main subject of the paper is the study of randomly stopped sums where n in (1) is replaced by a random variable (r.v.) $\nu$ taking values in ${\mathbb{N}}_0:=\{0,1,2,\cdots \}$. Throughout the paper, we assume that $\{{\xi }_1,{\xi }_2,\dots \}$ is a sequence of independent copies of r.v. $\xi$. In addition, we suppose that the generating r.v. $\xi$ is nonnegative, i.e., $\mathbb{P}({\xi }_1⩾0)=1$, and the counting r.v. $\nu$ is nondegenerate at zero, i.e., $\mathbb{P}(\nu ⩾1)>0$. We call such $\nu$ a counting random variable or a stopping moment and suppose throughout the paper that it is independent of the sequence $\{{\xi }_1,{\xi }_2,\dots \}$. According to our assumptions, r.v.s ${\xi }_1,{\xi }_2,\cdots$ are i.i.d. with common distribution function

In such a case, the distribution function (d.f.) of the randomly stopped sum has the form and the tail function (t.f.) of the randomly stopped sum for positive x has the form

The usual problem considered in many papers is to give conditions that guarantee that $F_{S_{\nu }}$ belongs to some regularity class, provided that d.f. $F_{\xi }$ or $F_{\nu }$ belongs to the regularity class under consideration. A detailed overview of regularity classes can be found in the books [1,2,3,4,5,6] and references therein, where all the main properties of those regularity classes are described and analyzed. In this paper, both in the main results and in the discussion of similar results, we essentially limit ourselves to five regularity classes: the class of long-tailed distributions $\mathcal{L}$, the class of distributions with dominantly varying tails $\mathcal{D}$, the class of distributions with consistently varying tails $\mathcal{C}$, the class of distributions with extended regularly varying tails $\mathcal{ERV}$, and the class of regularly varying distributions $\mathcal{R}$. Here, we only note that all the mentioned regularity classes of distributions are subclasses of the heavy-tailed distributions $\mathcal{H}$. The rest of the article is organized as follows. Section 2 is intended to introduce the classes of distributions under consideration. This section provides precise definitions of those classes, discusses the main properties of distributions from the classes, and presents several examples of distributions. In Section 3, we present the two results that motivated us to write this article. The formulated theorems determine under what conditions a randomly stopped sum’s regularity induces the stopping moment’s regularity. In Section 4, we present two theorems and two propositions, in which we formulate the main results of our paper. Section 5 of our paper is devoted to proofs. In it, we present the auxiliary lemmas used and provide detailed proof of the main results. In Section 6, we present an example showing how a randomly stopped sum’s regularity affects the stopping moment’s regularity. The asymptotic formulas established in that example are entirely consistent with the main results of the article. The last Section 7 is devoted to the discussion of the obtained results.

In this section, we present definitions of the regularity classes under consideration and discuss known results related to the closure of these classes under random stopped sums. As mentioned earlier, in this paper, we only study the properties of nonnegative random variable distributions. The results presented below are undoubtedly correct for nonnegative random variable distributions, whereas the results presented for distributions of r.v.s that can take negative values can be incorrect. However, all the definitions below can be applied to the distributions of r.v.s with both positive and negative values. We begin with the narrowest class of distributions $\mathcal{R}$.

•A d.f. F with t.f. $\overline{F}=1-F$ is said to be regularly varying with index $\alpha ⩾0$, denoted $F\in {\mathcal{R}}_{\alpha }$, if for all $y>0$,

The class of all regularly varying distributions is

It is evident that d.f.s are regularly varying.

Problems related to the randomly stopped sums of regularly varying d.f.s are considered in [7,8,9,10,21], among others. For instance, Proposition 4.1 in [10] states that

Here and further, the notation $a\left(x\right)\underset{x\to \infty }{\sim }b\left(x\right)$ for positive functions $a\left(x\right)$ and $b\left(x\right)$ means that and the notation $a\left(x\right)\underset{x\to \infty }{\lesssim }b\left(x\right)$ for two positive functions $a\left(x\right)$ and $b\left(x\right)$ means that

The direct generalization of the class $\mathcal{R}$ is the class $\mathcal{ERV}$ introduced in [11] and further considered in [1,12,13,14,15,16,17,18,19].

•A d.f. F is said to be extended regularly varying with indices $0⩽\alpha ⩽\beta <\infty$, denoted $F\in {\mathcal{ERV}}_{\{\alpha ,\beta \}}$, if for all $y>1$,

The class of all extended regularly varying distributions is

Since $\overline{F}\left(xy\right)⩽\overline{F}\left(x\right)$ for any d.f. F and all $x>0$ and $y⩾1$, the distribution F belongs to the class $\mathcal{ERV}$ if for all $y>1$ and some $\beta <\infty$; see [20].

Another somewhat broader class of distributions is the class of distributions with consistently varying tails $\mathcal{C}$.

•A d.f. F is said to have a consistently varying tail, denoted $F\in \mathcal{C}$, if

The definitions presented imply that $\mathcal{R}\subset \mathcal{ERV}\subset \mathcal{C}$. Therefore all d.f.s $F_1,F_2,F_3$, and $F_4$, presented above, have consistently varying tails. Due to the considerations in [22,23,24], d.f. belongs to the class $\mathcal{C}$ for any parameter $p\in (0,1)$, where $\lfloor a\rfloor$ is the integer part of $a\in \mathbb{R}$. Namely, for $x\in \left[2^n,2^{n+1}\right)$ and $1/2<y⩽1$, we have that $\lfloor {log}_{2}x\rfloor =n$ and $\lfloor {log}_{2}xy\rfloor \in \{n-1,n\}$. If $\lfloor {log}_{2}x\rfloor =\lfloor {log}_{2}xy\rfloor =n$, then

If $\lfloor {log}_{2}x\rfloor =n$ and $\lfloor {log}_{2}xy\rfloor =n-1$, then

It is clear that the derived inequalities imply that $F_5\in \mathcal{C}$. We also note that both inclusions in the relation $\mathcal{R}\subset \mathcal{ERV}\subset \mathcal{C}$ are proper. Due to the results of [20], the distribution with t.f. belongs to $\mathcal{ERV}\setminus \mathcal{R}$, and the distribution with t.f. belongs to $\mathcal{C}\setminus \mathcal{ERV}$.

Problems related to the randomly stopped sums of d.f.s from class $\mathcal{C}$ are considered in [22,25,26,27], among others. For instance, [26] states the following: where is the upper Matuszewska index of d.f. $F_{\xi }$; see [28,29].

Another, even broader class is the class of distributions with dominatedly varying tails $\mathcal{D}$, described more than half a century ago in [30].

•A d.f. F is said to have a dominatedly varying tail, denoted $F\in \mathcal{D}$, if

It is obvious that $\mathcal{R}\subset \mathcal{ERV}\subset \mathcal{C}\subset \mathcal{D}$ and that $F\in \mathcal{D}\iff J_F^{+}<\infty$. Hence all the above-mentioned d.f.s $F_1,F_2,F_3,F_4,F_5,F_6$, and $F_7$ belong to the class $\mathcal{D}$. As noted in [31], the Peter and Paul distribution is an example of a distribution belonging to the class $\mathcal{D}$ but not to the class $\mathcal{C}$. We recall that r.v. $\xi$ is said to be distributed according to the generalized Peter and Paul distribution with parameters $a>0$ and $b>1$ if its t.f. for $x⩾1$ has the following expression:

For this distribution, we derive (for details, see [23]) that which implies $F_{\xi }\in \mathcal{D}\setminus \mathcal{C}$.

The asymptotic properties of randomly stopped sums with summands that vary dominatedly are considered in [32,33,34,35,36]. For instance, [33] Theorem 5] states that

Class $\mathcal{C}$ can be extended in the other direction instead of distributions with dominatedly varying tails $\mathcal{D}$ by choosing the class of long-tailed distributions $\mathcal{L}$. The class of long-tailed distributions was introduced by Chistyakov [37] in the context of branching processes and became one of the most important subclasses of heavy-tailed distributions.

•A distribution F is said to belong to a class of long-tailed distributions $\mathcal{L}$ if

for all (equivalently, for some) $y\in \mathbb{R}$.

However, all the definitions below can be applied to distributions of random variables taking on both positive and negative values.

According to the above considerations, $\mathcal{R}\subset \mathcal{ERV}\subset \mathcal{C}\subset L\cap \mathcal{D}\subset \mathcal{L}$ with all proper inclusions. To see that $\mathcal{L}\cap \mathcal{D}\setminus \mathcal{C}\ne \varnothing$, we can take d.f. $F_9$ from [20] with t.f. and a simple Weibull distribution with d.f. shows that inclusion $\mathcal{L}\cap \mathcal{D}\subset \mathcal{L}$ is also proper.

The asymptotic properties of randomly stopped sums with summands with long-tailed distributions are considered in [33,36,38,39,40,41,42,43]. Asymptotic properties in the above papers are found either exclusively for summands with long tails or for summands satisfying additional conditions. For instance, [40] Theorem 8] states that

It was mentioned earlier that all discussed regularity classes consist of heavy-tailed distributions.

•A d.f. F is said to be heavy-tailed, denoted by $F\in \mathcal{H}$, if

Due to the inclusions $\mathcal{R}\subset \mathcal{ERV}\subset \mathcal{C}\subset \mathcal{L}\cap \mathcal{D}\subset \mathcal{L}$, all the participating classes are heavy-tailed because $\mathcal{L}\subset \mathcal{H}$. This fact can be easily observed because of the property

(see, e.g., comments in [44]) and the criteria

(see, e.g., [45] Lemma 1]).

The randomly stopped sum with heavy-tailed nonnegative i.i.d. summands $\{{\xi }_1,{\xi }_2,\dots \}$ satisfies the following simple estimate:

Hence we have the following statement for any counting r.v. $\nu$:

In addition, for any positive $\lambda$ and i.i.d. summands $\{{\xi }_1,{\xi }_2,\dots \}$, we have

This leads to the following statement for any nondegenerate at zero distribution $F_{\xi }$ of $\{{\xi }_1,{\xi }_2,\dots \}$:

In Section 2, we described the results of direct type on the regularity of the randomly stopped sums. In all the mentioned results, we specified the conditions on random summands and counting r.v. ensuring that randomly stopped sums fall into the desired regularity class. However, the paper [10] also presents results of the inverse type. Namely, Proposition 4.9 of [10] consists of the following two statements.

In this paper, we prove analogous statements in the broader classes of distributions $\mathcal{ERV}$ and $\mathcal{D}$. We present exact formulations of the statements in Section 4.

As mentioned above, in this section, we formulate the main results of the work on how the membership of the random stopped sum distribution in classes $\mathcal{ERV}$ and $\mathcal{D}$ affects the membership of the counting r.v. distribution in the class $\mathcal{D}$. If the randomly stopped sum distribution belongs to the class $\mathcal{ERV}$, then we also obtain asymptotic formulas similar to those in Theorems 1 and 2. Our first statement is an analog of Theorem 1 for the wider regularity class $\mathcal{ERV}$.

It is evident that Theorem 3 implies the statement of Theorem 1 because $\alpha =\beta$ in the case where d.f. $F_{\xi }$ of r.v. $\xi$ belongs to the class ${\mathcal{R}}_{\alpha }$.

Just as Theorem 3 generalizes Theorem 1, the following theorem generalizes Theorem 2 in a similar way.

Both theorems can be proved based on the propositions below. Note that these two propositions are interesting in themselves and that we prove them in a completely different way compared to the proofs of Theorems 1 and 2 given in [10]. In addition, we observe that Propositions 2 and 3 present statements analogous to Theorems 1 and 2 but for class $\mathcal{D}$ instead of $\mathcal{R}$. Section 5 presents detailed proofs of the theorems and propositions below.

In this section, we present proofs of the propositions and theorems stated in Section 4.