Randomly stopped maximum and maximum of sums with consistently varying distributions

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=\xi_1+\xi_2+\cdots+\xi_n$ for $n\geqslant1$. We consider conditions for random variables $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution functions of the random maximum $\xi_{(\eta)}:=\max\{0,\xi_1,\xi_2,\ldots,\xi_{\eta}\}$ and of the random maximum of sums $S_{(\eta)}:=\max\{S_0,S_1,S_2,\ldots,S_{\eta}\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.

Before discussing the properties of F ξ (η) and F S (η) , we recall the definitions of some classes of heavy-tailed d.f.s. For a d.f. F , we denote F (x) = 1 − F (x) for real x.
If a d.f. G is supported on R, then we say that G is subexponential It is known (see, e.g., [5,12,15], and Chapters 1.4 and A3 in [10]) that these classes satisfy the following inclusions: These inclusions are depicted in Fig. 1 borrowed from the paper [14]. In this figure, the class C of distributions having consistently varying tails is highlighted.
It should be noted that the subject of the paper is partially motivated by the closure problem of the random convolution. In the case of independent and identically distributed (i.i.d.) r.v.s {ξ, ξ 1 , ξ 2 , . . .}, we say that a class K of d.f.s is closed with respect to the random convolution if the condition F ξ ∈ K implies F Sη ∈ K. The first result on the convolution closure of subexponential distributions was obtained by Embrechts and Goldie (see Thm. 4.2 in [11]) and by Cline (see Thm. 2.13 in [6]). The random closure results for class D can be found in [7,16], and for the class L, in [1,16,19,20]. The random closure results for the class C can be derived from the results of [14]. We note that in [7,14,20], the case of not necessarily identically In this work, we consider the randomly stopped maximum ξ (η) and randomly stopped maximum of sums S (η) of independent but not necessarily identically dis-tributed r.v.s. As was noted before, we restrict our consideration to the class C but extend it to the real-valued r.v.s as in Theorem 3.
If r.v.s {ξ 1 , ξ 2 , . . .} are not identically distributed, then different collections of conditions on r.v.s {ξ 1 , ξ 2 , . . .} and η imply that F ξ (η) ∈ C or F S (η) ∈ C. We suppose that some r.v.s from {ξ 1 , ξ 2 , . . .} have distributions belonging to the class C, and we find conditions for r.v.s {ξ 1 , ξ 2 , . . .} and η such that the distribution of the randomly stopped maximum or the randomly stopped maximum of sums remains in the same class. The results presented and their proofs are closely related to the results of the papers [7,8,14].
It is worth noting that the closure properties for d.f.s F S (η) in the case of i.i.d. r.v.s can be derived, for instance, from the asymptotic formulas obtained in [9,13,17,18,21]. Unfortunately, in the case of nonidentically distributed r.v.s, similar asymptotic formulas do not exist. Therefore we have to use other methods to prove our main results.
The rest of the paper is organized as follows. In Section 2, we present our main results together with a few examples of randomly stopped maximum ξ (η) and randomly stopped maximum of sums S (η) with d.f.s having consistently varying tails. Section 3 is a collection of auxiliary lemmas, and the proofs of the main results are presented in Section 4.

Main results
In this section, we present four statements, two theorems and two corollaries. Theorem 4 and Corollary 1 deal with the belonging of the d.f. F ξ (η) to the class C. (a) F ξκ ∈ C for some κ ∈ supp(η), (a) F ξ k ∈ C for each k ∈ N, Similarly to Corollary 1, we can state the following corollary. We note that in the • r.v.s ξ k are exponentially distributed for k ≡ 0 mod 3, that is, • r.v.s ξ k are degenerate at zero for all k ≡ 2 mod 3 and k ≡ 1 mod 3, k 4, that is, • ξ 1 = (1+U)2 G , where r.v.s U and G are independent, U is uniformly distributed on the interval [0, 1], and G is geometrically distributed with parameter q ∈ (0, 1), that is, In addition, let η be a counting r.v. independent of {ξ 1 , ξ 2 , . . .}.
Theorem 5 implies that the d.f. of the randomly stopped maximum of sums S (η) belongs to the class C because:

Auxiliary lemmas
In this section, we give auxiliary lemmas, which we use in the proof of Theorem 5. The first lemma was proved in [17, Thm. 2.1]; a more general case can be found in [3,Thm. 2.1]. We recall only that C ⊂ L. Hence, the statement of the next lemma holds for d.f.s with consistently varying tails.
The second auxiliary lemma was proved in [14,Lemma 3]. It describes the situation where the d.f. of sums of independent r.v.s belongs to the class C. Lemma 2. Let {X 1 , X 2 , . . . , X n } be independent real-valued r.v.s. The d.f. of the sum Σ n := X 1 + X 2 + · · ·+ X n belongs to the class C if the following two conditions are satisfied: The following statement was proved in [7,Lemma 3.2]. It gives an upper estimate for the tail of the sum of r.v.s from the class D.
Then, for each p > J + FX ν , there exists a positive constant c 1 such that for all n ν and x 0, where F Σn is the d.f. of the sum Σ n = X 1 + · · · + X n . The last useful lemma is an obvious conclusion from Theorem 3.1 in [4].

Proofs of the main results
Proof of Theorem 4. It suffices to prove that For all x > 0 and K ∈ N, we have {ξ k > x} P(η = n). Therefore, for all x > 0 and y ∈ (0, 1). Denote If x > 0, 1/2 y < 1, and K κ, then because and F ξκ ∈ C ⊂ D.
The obtained asymptotic relation (5) and estimate (4) imply For each 1 n K, we have due to the Bonferroni inequality. This implies that, for an arbitrary ε > 0, if x is sufficiently large. Substituting the last estimate into inequality (6), we get because F ξ k ∈ C for each k / ∈ K. Since ε > 0 is arbitrary, the last estimate implies that lim sup y↑1 lim sup x→∞ J 1 1.
According to Lemma 2, the d.f. F Sn belongs to the class C for each fixed n. Therefore, lim sup y↑1 lim sup x→∞ I 1 1.
According to the conditions of the theorem, the d.f. F ξ k belongs to the class C for each fixed index k ∈ N. Hence, using Lemma 4, we get The last two estimates imply that for sufficiently large x. Therefore, by inequalities (12) and (13) Finally, substituting estimates (11) and (14) into (10), we obtain that lim sup y↑1 lim sup x→∞ P(S (η) > xy) P(S (η) > x) 1 + 2c 3 P(η = a) E η p+1 1 [K+1,∞) (η) , for an arbitrary K a. This inequality implies the desired relation (9) due to condition (c) of the theorem. Theorem 5 is proved.