Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{\xi_1,\xi_2,\ldots\}$ 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 $\mathcal{O}$-exponential distributions.

We are interested in conditions under which the d.f. of S η F Sη (x) = P(S η x) = ∞ n=0 P(η = n)P(S n x) belongs to the class of O-exponential distributions. According to Albin and Sunden [1] or Shimura and Watanabe [15], a d.f. F belongs to the class of O-exponential distributions OL if or, equivalently, if and only if The last condition shows that class OL is quite wide. We further describe some more popular subclasses of OL for which we will present some results on the random convolution of distributions from these subclasses.
A d.f. F is said to belong to the class L of long-tailed d.f.s if for every fixed a > 0, we have F (x) = 1.
A d.f. F is said to belong to the class L(γ) of exponential distributions with some γ > 0 if for any fixed a > 0, we have A d.f. F belongs to the class D (or has a dominatingly varying tail) if for every fixed a ∈ (0, 1), we have where, as usual, * denotes the convolution of d.f.s.
A d.f. F supported on the interval [0, ∞) belongs to the class S * ( or is strongly subexponential) if If a d.f. F is supported on R, then F belongs to some of the classes S or S * if The presented definitions, together with Lemma 2 of Chistyakov [2], Lemma 9 of Denisov et al. [5], Lemma 1.3.5(a) of Embrechts et al. [9], and Lemma 1 of Kaas and Tang [11], imply that Now we present a few known results on when the d.f. F Sη belongs to some class. The first result about subexponential distributions was proved by Embrechts and Goldie (Theorem 4.2 in [8]) and Cline (Theorem 2.13 in [3]).
In the case of strongly subexponential d.f.s, the following result, which involves weaker restrictions on the r.v. η, can be derived from Theorem 1 of Denisov et al. [6] and Corollary 2.36 of Foss et al. [10]. Theorem 2. Let {ξ 1 , ξ 2 , . . .} be independent copies of a nonnegative r.v. ξ with strongly subexponential d.f. F ξ and finite mean Eξ. Let η be a counting r.v. independent of Similar results for classes D, L, and OL can be found in the papers of Leipus and Šiaulys [12] and Danilenko and Šiaulys [4]. We further present Theorem 6 from [12].
In all presented results, r.v.s {ξ 1 , ξ 2 , . . .} are identically distributed. In this work, we consider independent, but not necessarily identically distributed, r.v.s. As was noted, we restrict our consideration on the class OL. In fact, in this paper, we generalize the results of [4]. If {ξ 1 , ξ 2 , . . .} may be not identically distributed, then various collections of conditions on r.v.s {ξ 1 , ξ 2 , . . .} and η imply that F Sη ∈ OL. The rest of the paper is organized as follows. In Section 2, we formulate our main results. In Section 3, we present all auxiliary assertions, and the detailed proofs of the main results are presented in Section 4. Finally, a few examples of O-exponential random sums are described in Section 5.

Main results
In this section, we formulate our main results. The first result describes the situation where the tails of d.f.s F ξ k for large indices k are uniformly comparable with itself at the points x and x − 1 for all x ∈ [0, ∞). • For some κ ∈ supp(η) \ {0} = {n ∈ N : P(η = n) > 0}, F ξκ ∈ OL.
• For each k ∈ supp(η), k κ, either lim Since each d.f. from the class OL is comparable with itself, the next assertion follows immediately from Theorem 4.
Our second main assertion is dealt with counting r.v.s having finite support.
Our last main assertion describes the case where the tails of d.f.s F ξ k are comparable at x and x − 1 asymptotically and uniformly with respect to large indices k. In this case, conditions are more restrictive for a counting r.v.

Auxiliary lemmas
In this section, we present all assertions that we use in the proofs of our main results. We present some of auxiliary results with proofs. The first assertion can be found in [7] (see Eq. (2.12)).
for all x ∈ R, v ∈ R, and t > 0.
The following assertion is the well-known Kolmogorov-Rogozin inequality for concentration functions. Recall that the Lévy concentration function or simply concentration function of a r.v. X is the function The proof of the next lemma can be found in [14] (Theorem 2.15). Lemma 2. Let X 1 , X 2 , . . . , X n be independent r.v.s, and let Z n = n k=1 X k . Then, for all n ∈ N, where A is an absolute constant, and 0 < λ k λ for each k ∈ {1, 2, . . . , n}.
The following assertion describes sufficient conditions under which the d.f. of two independent r.v.s belongs to the class OL.
and one of the following two conditions holds: • F X2 ∈ OL.
Proof. We split the proof into three parts. I. First, suppose that P(X 2 D) = 1 for some D > 0. In this case, condition (2) holds evidently.
For each real x, we have Hence, for such x, This estimate implies that II. Now let us consider the case where condition (2) holds but F X2 (x) > 0 for all x ∈ R. For each real x, we have Therefore, for all M, x such that 0 < M < x − 1. In addition, for such M and x, we obtain The obtained estimates imply that for all x and M such that 0 < M < x − 1. Consequently, for all positive M . Therefore, (1). III. It remains to prove the assertion when both d.f.s F X1 and F X2 are O-exponential. By Lemma 1 we have for all x and M such that 0 < M < x − 1. Therefore, for every positive M , Letting M tend to infinity, we get that because F X1 and F X2 belong to class OL. Consequently, F X1 * F X2 ∈ OL due to requirement (1). Lemma 3 is proved.
Proof. We use induction on n. If n = 2, then the statement follows from Lemma 3. Suppose that the statement holds if n = m, that is, F X1 * F X2 * · · · * F Xm ∈ OL, and we will show that the statement is correct for n = m + 1.
Conditions of the lemma imply that F Xm+1 ∈ OL or So, using Lemma 3 again, we get We see that the statement of the lemma holds for n = m + 1 and, consequently, by induction, for all n ∈ N. The lemma is proved.

Proofs of the main results
In this section, we present proofs of our main results.
Proof of Theorem 4. Conditions of Theorem and Lemma 4 imply that the d.f. F Sκ (x) = P(S κ x) belongs to the class OL. So, we have or, equivalently, for some positive constant c 1 . We observe that, for all x 0, where Since κ ∈ supp(η), we obtain Hence, it follows from (3) that By Lemma 1 we have for all real x and M . The third condition of the theorem implies that for all k ∈ N and some positive c 2 .
If we choose M = x/2 in estimate (7), then, using (4), we get Applying Lemma 1 again, we obtain By choosing M = x/2 we get from inequalities (8) and (9) that Continuing the process, we find for all k ∈ N. Therefore, for all x 0.
Proof of Theorem 5. The statement of the theorem can be derived from Theorem 4 or proved directly. We present the direct proof of Theorem 5. It is evident that S k = ξ κ + k n=1, n =κ ξ n for each k κ. Hence, by Lemma 4, F S k ∈ OL for all κ k D.
If x 1, then we have where in the last step we use the inequality a 1 + a 2 + · · · + a n b 1 + b 2 + · · · + b n max a 1 b 1 , a 2 b 2 , . . . , a n b n , provided that n 1 and a i , b i > 0 for i ∈ {1, 2, . . . , n}.
Since F Sn ∈ OL for all n κ, we get from (11) that and the statement of Theorem 5 follows.
Proof of Theorem 6. As usual, it suffices to prove relation (12). If x 0, then we have Similarly, for K 2 and x 2K, The distribution function F Sκ belongs to the class OL due to Lemma 4. So, by estimate (6) we have Now we consider the sum K 2 (x). Since F Sκ is O-exponential, we have with some positive constant c 4 . On the other hand, the third condition of Theorem 6 implies that for some constants c 5 > 2, c 6 > 0 and all k ∈ N.
By Lemma 1 (with v = c 5 ) we have Consequently, Applying Lemma 1 again for the sum S κ+2 = S κ+1 + ξ κ+2 (with v = x/2 + 1/2), we get If x 2(c 5 − 1) + 1, then x/2 + 1/2 c 5 . Therefore, by the last inequality we obtain that Applying Lemma 1 once again (with v = x/3 + 2/3), we get If x 3(c 5 − 1) + 1, then 2x/3 + 1/3 2(c 5 − 1) + 1 and x/3 + 2/3 c 5 . So, the last estimate implies Continuing the process, we can get that for all k ∈ N. We can suppose that K = c 5 in representation (14). In such a case, it follows from inequality (16) that Since, obviously, it remains to estimate sum K 3 (x). Using Lemma 2, we obtain with some absolute positive constant A. By the fourth condition of the theorem, for some 0 < ∆ < 1 and all sufficiently large k. So, for such k, From the last estimate it follows that for sufficiently large x. Therefore, by estimate (13) and the last condition of the theorem. Representation (14) and estimates (15), (17) where k ∈ {1, 2, . . . , D}, D 1, and α > 0. In addition, we suppose that the r.v. ξ D+k for each k ∈ N is distributed according to the exponential law with parameter λ/k, that is, F ξ D+k (x) = e −λx/k , x 0.
Here the last estimate is the well-known Chernof bound for the Poisson law (see, e.g., p. 97 in [13]).
As we can see, the r.v.s {ξ 1 , ξ 2 , . . .} from the last example satisfy the conditions of Theorem 6, whereas the third condition of Theorem 4 does not hold because, in this case,