Randomly stopped sums with exponential-type distributions

Abstract. Assume that {ξ1, ξ2, . . .} are independent and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at zero and integer-valued random variable, which is independent of {ξ1, ξ2, . . .}. In this paper, we consider conditions for η and {ξ1, ξ2, . . .} under which the distribution of the random sum ξ1+ ξ2+ · · ·+ ξη belongs to the class of exponential distributions.


S. Danilenko et al.
In this paper, we consider possibly nonidentically distributed r.v.s ξ 1 , ξ 2 , . . . .We find conditions under which d.f.F Sη belongs to the class of exponential distributions.If F ξ1 , F ξ2 , . . .are different, then the various collections of conditions on d.f.s {F ξ1 , F ξ2 , . ..}, and the counting r.v.η imply exponentiality of d.f.F Sη .Before discussing the properties of F Sη , we recall some d.f.classes related to exponentiality.
• For γ > 0, by L(γ) we denote the class of exponential d.f.s.It is said that F ∈ L(γ) if for any y > 0, lim x→∞ F (x + y) F (x) = e −γy .
• For γ = 0, the class L(0) is called the long-tailed distribution class and is denoted by L.
or, equivalently, According to Proposition 2.6 by Albin and Sundén in [2], an absolutely continuous d.f.F belongs to the class L(γ) if and only if for some measurable functions α and β with α(u) + β(u) 0, for all u ∈ R such that exists.We note that each exponential distribution, each Erlang's distribution and each gamma distribution belong to the class L(γ) with some γ > 0.
It is easy to see that the following two inclusions hold: In [4,5], Cline claimed that d.f.F Sη belongs to the class L(γ) for some γ 0 if r.v.s {ξ 1 , ξ 2 , . ..} are identically distributed with d.f.F ξ ∈ L(γ) and η is any counting r.v.Albin [1] constructed a counterexample and showed that Cline's result is false in general.In his paper [1], Albin stated that d.f.F Sη remains in the class L(γ) for some γ 0 if r.v.s {ξ 1 , ξ 2 , . ..} are identically distributed with common d.f.F ξ belonging to the class L(γ) and Ee δη < ∞ for each δ > 0. In order to prove his statement, Albin used the following implication for c ∈ R: provided that ε > 0, t ∈ R and F is a d.f.from the class L(γ) for some γ 0.Here and later F * n denotes n-fold convolution of d.f.F with itself.Unfortunately, if γ > 0, then the obtained relation holds for positive t only.Watanabe and Yamamuro (see [13,Remark 6.1]) showed that the above implication is incorrect in the case of positive γ and negative t.When γ = 0, the above implication for positive t is sufficient to prove the Albin's statement under the weaker restrictions on the counting r.v.η (see [11,Thm. 6]).The Albin's statement on conditions for which F Sη ∈ L(γ) has remained only as a hypothesis in the case γ > 0. Watanabe and Yamamuro [13] do not prove the Albin's hypothesis in this case, they presented the following statement (see [13,Prop. 6.1]).
Theorem 2. Let {ξ 1 , ξ 2 , . ..} be a sequence of independent nonnegative r.v.s with a common d.f.F ξ such that F * κ ξ ∈ L(γ) for some integer κ 1 and some γ 0. In addition, let counting r.v.η be independent of {ξ 1 , ξ 2 , . ..} and P(η κ) > 0. Then F Sη ∈ L(γ) if any pair (i), (ii) or (i), (iii) of conditions holds, where (i) for any ε ∈ (0, 1), there is an integer M = M (ε) such that for x 0, Motivated by the presented results, we also consider conditions for which d.f.F Sη belongs to the class L(γ) for some γ 0.Here the randomly stopped sum S η contains independent but not necessarily identically distributed r.v.s.We suppose that some d.f.s from {F ξ1 , F ξ2 , . ..} belongs to the exponential class, and we find conditions for F ξ1 , F ξ2 , . . .and η such that the distribution of the randomly stopped sum S η remains in the same class.In this work, we present three collections of such conditions.The proofs of the main results are based on ideas from the papers [6-8, 10, 13] and [15].The similar results for class L = L(0) were obtained in the papers [12] and [14].
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 having some exponential distributions.Section 3 is a collection of auxiliary lemmas.The proofs of the three main results are presented in Section 4.

Main results and examples
At first, in this section, we present three theorems, which deal with situations when the d.f.F Sη belongs to the class L(γ) for some γ 0. In Theorem 3, the case of a finitely supported counting r.v.η is considered, while Theorems 4 and 5 deal with the case of unbounded right tail of η.In Theorems 3 and 5, we consider nonnegative r.v.s, while in Theorem 4, r.v.s ξ 1 , ξ 2 , . . .can be real valued.
Here we observe that condition (1) is equivalent to the two-sided estimate which holds for some γ 0 and for each fixed y 0.
In addition, we observe that condition (2) implies that P(η = k) > 0 for all sufficiently large k.
Theorem 5. Let {ξ 1 , ξ 2 , . ..} be a sequence of nonnegative independent r.v.s with d.f.s {F ξ1 , F ξ2 , . ..}, and let η be a counting r.v.independent of {ξ 1 , ξ 2 , . ..}. D.f.F Sη belongs to the class L(γ) for some γ 0 if there exist κ 1 and 1 ν κ such that Further in this section, we present two examples, which illustrate several applications of our theorems.In both examples, we construct randomly stopped sums that belong to the class of exponential distributions.
Example 1. Suppose that we have a three-seasonal sequence of independent Erlang r.v.s with d.f.s from class L(2), i.e.
It is clear that We see that all conditions of Theorem 3 are satisfied.Consequently, d.f.F Sη ∈ L(2).
Consequently, d.f.F Sη belongs to the class L(1) due to assertion of Theorem 5.

Auxiliary lemmas
In this section, we give all auxiliary assertions, which we use in the proofs of our main results.The first lemma was proved by Embrechts and Goldie (see [9,Thm. 3]).
Lemma 1.Let F and G be two d.f.s, and let F belong to the class L(γ) for some γ 0.
Then convolution F * G belongs to the class L(γ) if one of the following conditions holds: The second lemma is the inhomogeneous case of the upper estimate, which was presented in the proof of Proposition 6.1 from [13].
https://www.mii.vu.lt/NAProof.For any x and any b > 0, we have Condition (3) implies that for any fixed ε ∈ (0, 1) if y x + a − b (then x − y b − a) and b is sufficiently large.
For such b, we get On the other hand, it is obvious that Therefore, for any ε ∈ (0, 1) and sufficiently large b = b(a, ε), we obtain Lemma 2 is proved.
Proof.Due to representation (4), we have for arbitrary real x and positive b.
According to (5), for fixed ε ∈ (0, 1/2), we have Similarly as in the proof of Lemma 2, for such b, we get Therefore, and the assertion of Lemma 3 follows. https://www.mii.vu.lt/NA The last auxiliary assertion is a mild generalization of Braverman's lemma (see [3,Lemma 1]).Lemma 4. Let {ξ 1 , ξ 2 , . ..} be independent r.v.s, and let F ξ1 ∈ L(γ) for some γ 0.Then, for each a > 0, there exists a constant c a such that for all n ∈ N and x ∈ R.
Proof.The definition of the class L(γ) implies that if x > x a and x a is sufficiently large.
If x x a , then, obviously, . Consequently, If n 2, then, for any x ∈ R, we have So, the assertion of Lemma 4 follows.

Proofs of main results
In this section, we present detailed proofs of all our main results.For these proofs, we use essentially approaches from [6,10] and [13].
Proof of Theorem 3.For each positive x, we have Since support supp η is finite, for an arbitrary positive y, we have due to the two-sided inequality Since F ξν ∈ L(γ) for some 1 ν min{supp η \ {0}}, the set of indices K is not empty.Lemma 1 implies that d.f.F K of sum i∈K ξ i belongs to the class L(γ).
Further, if i * / ∈ K, then F ξ i * (x) = o(F ξν (x)) because of the theorem's conditions.Therefore, and consequently, F K * F ξ i * belongs to the class L(γ) due to the second part of Lemma 1.
Continuing our considerations, we get that d.f.
belongs to the class L(γ) as well for an arbitrary index k ∈ supp η.Here * i / ∈K F ξi denotes d.f. of sum i / ∈K ξ i .Consequently, the double estimate (6) implies the following two inequalities: for each positive y.The last two estimates finish the proof of Theorem 3.
https://www.mii.vu.lt/NAProof of Theorem 4. In order to show that F Sη ∈ L(γ) for some γ 0, it is sufficient to derive the following two estimates: which both should valid for each positive y.
(I) At first, we show inequality (7).For this, we suppose that y is an arbitrary positive number, and we choose ε ∈ (0, 1).According to condition (2), we have for all n N = N (ε) 2. For such N , we get that Using Lemma 2, we obtain (1 + ε)e −γy P(η = n) F Sn (x) for some b = b(y, ε) > 0. This relation together with inequality (9) shows that Condition (1) implies that for all fixed k and u.It follows that F ξ k ∈ L(γ) for each k.Hence, according to Lemma 1, we obtain that F Sn ∈ L(γ) for each fixed n ∈ N. Therefore, if x x = x(N, y, ε).So, for the chosen N , b and for all x x, we have According to Lemma 4, Therefore, for all sufficiently large x.
The last inequality implies that Since ε ∈ (0, 1) is arbitrarily chosen, the desired inequality (7) holds for each positive y.
Letting ε tend to zero, from the last inequality we get the desired estimate (8).The theorem is proved.
Theorem 3 implies that F Sη ∈ L(γ) because of the finiteness of support supp η.The investigation analogous to that in part (II) implies that d.f.F S η belongs to the class L(γ) as well.Now the statement of theorem follows immediately from relation (16).Theorem 5 is proved.
for all y x + a − b and sufficiently large b = b(a, ε).