Randomly stopped 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. We consider conditions for $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.

We are interested in conditions under which the d.f. of S η , belongs to the class of consistently varying distributions.
Before discussing the properties of F S η , we recall the definitions of some classes of heavy-tailed d.f.s, where F (x) = 1 − F (x) for all real x and a d.f. F .
• A d.f. F is long-tailed (F ∈ L) if for every y (equivalently, for some y > 0), • A d.f. F has a dominatedly varying tail (F ∈ D) if for every fixed y ∈ (0, 1) (equivalently, for some y ∈ (0, 1)),  • It is known (see, e.g., [4,11,13], and Chapters 1.4 and A3 in [8]) that these classes satisfy the following inclusions: These inclusions are depicted in Fig. 1 with the class C highlighted. There exist many results on sufficient or necessary and sufficient conditions in order that the d.f. of the randomly stopped sum (1) belongs to some heavy-tailed distribution class. Here we present a few known results concerning the belonging of the d.f. F S η to some class. The first result on subexponential distributions was proved by Embrechts and Goldie (see Theorem 4.2 in [9]) and Cline (see Theorem 2.13 in [5]).
Similar results for the class D can be found in Leipus and Šiaulys [14]. We present the statement of Theorem 5 from this work.
The random convolution closure for the class L was considered, for instance, in [1,14,16,17]. We now present a particular statement of Theorem 1.1 from [17]. (i) P(η κ) > 0 for some κ ∈ N; (ii) for all k κ, the d.f. F S k of the sum S k is long tailed; We observe that the case of identically distributed r.v.s is considered in Theorems 1 and 2. In Theorem 3, r.v.s {ξ 1 , ξ 2 , . . .} are independent but not necessarily identically distributed. A similar result for r.v.s having d.f.s with dominatedly varying tails can be found in [6].
In this work, we consider randomly stopped sums of independent and not necessarily identically distributed r.v.s. As noted before, we restrict ourselves on the class C. If r.v.s {ξ 1 , ξ 2 , . . .} are not identically distributed, then different collections of conditions on {ξ 1 , ξ 2 , . . .} and η imply that 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 minimal conditions on {ξ 1 , ξ 2 , . . .} and η for the distribution of the randomly stopped sum S η to remain in the same class. It should be noted that we use the methods developed in [6] and [7].
The rest of the paper is organized as follows. In Section 2, we present our main results together with two examples of randomly stopped 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 three statements in which we describe the belonging of a randomly stopped sum to the class C. In the conditions of Theorem 5, the counting r.v. η has a finite support. Theorem 6 describes the situation where no moment conditions on the r.v.s {ξ 1 , ξ 2 , . . .} are required, but there is strict requirement for η. Theorem 7 deals with the opposite case: the r.v.s {ξ 1 , ξ 2 , . . .} should have finite means, whereas the requirement for η is weaker. It should be noted that the case of real-valued r.v.s {ξ 1 , ξ 2 , . . .} is considered in Theorems 5 and 6, whereas Theorem 7 deals with nonnegative r.v.s.
Then the d.f. F S η belongs to the class C if the following conditions are satisfied: Then the d.f. F S η belongs to the class C if the following conditions are satisfied: When {ξ 1 , ξ 2 , . . .} are identically distributed with common d.f. F ξ ∈ C, conditions (a), (b), and (c) of Theorem 6 are satisfied obviously. Hence, we have the following corollary.
Similarly to Corollary 1, we can formulate the following statement. We note that, in the i.i.d. case, conditions (a), (b), (e), and (f) of Theorem 7 are satisfied.
Then the d.f. F S η belongs to the class C under the following two conditions: Further in this section, we present two examples of r.v.s {ξ 1 , ξ 2 , . . .} and η for which the random sum F S η has a consistently varying tail. Example 1. Let {ξ 1 , ξ 2 , . . .} be independent r.v.s such that ξ k are exponentially distributed for all even k, that is, whereas, for each odd k, ξ k is a copy of the r.v.
where U and G are independent r.v.s, 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 , . . .} and distributed according to the Poisson law.
Theorem 6 implies that the d.f. of the randomly stopped sum S η belongs to the class C because: (a) F ξ 1 ∈ C due to considerations in pp. 122-123 of [2], (d) all moments of the r.v. η are finite.
Note that ξ 1 does not satisfy condition (c) of Theorem 7 in the case q 1/2. Hence, Example 1 describes the situation where Theorem 6 should be used instead of Theorem 7.

Example 2.
Let {ξ 1 , ξ 2 , . . .} be independent r.v.s such that ξ k are distributed according to the Pareto law (with tail index α = 2) for all odd k, and ξ k are exponentially distributed (with parameter equal to 1) for all even k, that is, In addition, let η be a counting r.v independent of {ξ 1 , ξ 2 , . . .} that has the Zeta distribution with parameter 4, that is, where ζ denotes the Riemann zeta function.
Theorem 7 implies that the d.f. of the randomly stopped sum S η belongs to the class C because: Regarding condition (d), it should be noted that the Zeta distribution with parameter 4 is a discrete version of Pareto distribution with tail index 3.
Note that η does not satisfy the condition (d) of Theorem 6 because J +

Auxiliary lemmas
This section deals with several auxiliary lemmas. The first lemma is Theorem 3.1 in [3] (see also Theorem 2.1 in [15] The following statement about nonnegative subexponential distributions was proved in Proposition 1 of [10] and later generalized to a wider distribution class in Corollary 3.19 of [12].
In the next lemma, we show in which cases the convolution F X 1 * F X 2 * · · · * F X n belongs to the class C. Lemma 3. Let {X 1 , X 2 , . . . , X n }, n ∈ N, be independent real-valued r.v.s. Then the d.f. F Σ n of the sum Σ n = X 1 + X 2 + · · · + X n belongs to the class C if the following conditions are satisfied: Proof. Evidently, we can suppose that n 2. We split our proof into two parts.
First part. Suppose that F X k ∈ C for all k ∈ {1, 2, . . . , n}. In such a case, the lemma follows from Lemma 1 and the inequality for a i 0 and b i > 0, i = 1, 2, . . . , m. Namely, using the relation of Lemma 1 and estimate (3) for arbitrary y ∈ (0, 1).
Since F X k ∈ C for each k, the last estimate implies that the d.f. F Σ n has a consistently varying tail, as desired.
Second part. Now suppose that F X k / ∈ C for some of indexes k ∈ {2, 3, . . . , n}. By the conditions of the lemma we have that F X k (x) = o(F X 1 (x)) for such k. Let K ⊂ {2, 3, . . . , n} be the subset of indexes k such that By Lemma 2, where Hence, lim sup x→∞ F Σ n (xy) for every y ∈ (0, 1). Equality (4) implies immediately that the d.f. F Σ n belongs to the class C. Therefore, the d.f. F Σ n also belongs to the class C according to the first part of the proof because Σ n = Σ n + k / ∈K X k and F X k ∈ C for each k / ∈ K. The lemma is proved.
The following two statements about dominatedly varying distributions are Lemma 3.2 and Lemma 3.3 in [6]. Since any consistently varying distribution is also dominatingly varying, these statements will be useful in the proofs of our main results concerning the class C.
Then, for each p > J + F Xν , there exists a positive constant c 1 such that for all n ν and x 0.

Proofs of the main results
Proof of Theorem 5. It suffices to prove that lim sup y↑1 lim sup x→∞ F S η (xy) According to estimate (3), for x > 0 and y ∈ (0, 1), we have Hence, by Lemma 3, which implies the desired estimate (6). The theorem is proved.
Proof of Theorem 6. As in Theorem 5, it suffices to prove inequality (6). For all K ∈ N and x > 0, we have Therefore, for x > 0 and y ∈ (0, 1), we have P(S η > xy) P(S η > x) = K n=1 P(S n > xy)P(η = n) P(S η > x) The random variable η is not degenerate at zero, so there exists a ∈ N such that P(η = a) > 0. If K a, then using inequality (3), we get Similarly as in the proof of Since C ⊂ D, we can use Lemma 4 for the numerator of J 2 to obtain ∞ n=K+1 P(S n > xy)P(η = n) c 3 F ξ 1 (xy) ∞ n=K+1 n p+1 P(η = n) with some positive constant c 3 . For the denominator of J 2 , we have that The conditions of the theorem imply that where F S a is the d.f. of the sum In addition, by Lemma 3 we have that the d.f. F S a belongs to the class C. If k / ∈ K a , then F ξ k ∈ C by the conditions of the theorem. This fact and Lemma 1 imply that lim inf x→∞ P(S a > x) Hence, for x sufficiently large. Therefore, Estimates (7), (8), and (10) imply that lim sup y↑1 lim sup x→∞ P(S η > xy) Eη p+1 1 {η>K} for arbitrary K a.
Letting K tend to infinity, we get the desired estimate (6) due to condition (d). The theorem is proved.
Proof of Theorem 7. Once again, it suffices to prove inequality (6).
By condition (e) we have that there exist two positive constants c 4 and c 5 such that Therefore, for a positive constant c 6 and all n ∈ N.
Using inequality (15)  for K a.
Letting K tend to infinity, we get the desired estimate (6) because Eη < ∞ by conditions (c) and (d). The theorem is proved.