Randomly Stopped Sums with Generalized Subexponential Distribution

1. Introduction

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent random variables (r.v.s) with distribution functions (d.f.s) $\{F_{{\xi }_1},F_{{\xi }_2},\dots \}$, and let $\eta$ be a counting random variable, that is, a nonnegative, nondegenerate at 0, and integer-valued r.v. In addition, we suppose that the r.v. $\eta$ and the sequence $\{{\xi }_1,{\xi }_2,\dots \}$ are independent.

Let $S_0:=0$, $S_n:={\xi }_1+\dots +{\xi }_n$ for $n\in \mathbb{N}$, and let

$$S_{\eta }=\sum _{k=1}^{\eta }{\xi }_k$$

be the randomly stopped sum of the r.v.s ${\xi }_1,{\xi }_2,\dots$.

By $F_{S_{\eta }}$ we denote the d.f. of $S_{\eta }$, and by $\overline{F}$ we denote the tail function (t.f.) of a d.f. F, that is, $\overline{F}\left(x\right)=1-F\left(x\right)$ for $x\in \mathbb{R}$. It is obvious that the following equalities hold for positive x:

$$\begin{array}{cc}\hfill F_{S_{\eta }}\left(x\right)& =\mathbb{P}(\eta =0)+\sum _{n=1}^{\infty }\mathbb{P}(\eta =n)\mathbb{P}(S_n⩽x)\hfill \\ \hfill {\overline{F}}_{S_{\eta }}\left(x\right)& =\sum _{n=1}^{\infty }\mathbb{P}(\eta =n)\mathbb{P}(S_n>x).\hfill \end{array}$$

In this paper, we consider a sequence $\{{\xi }_1,{\xi }_2,\dots \}$ of independent and possibly nonidentically distributed r.v.s. We suppose that some of d.f.s of these r.v.s belong to the class of generalized subexponential distributions $\mathcal{OS}$, and we find conditions under which d.f. $F_{S_{\eta }}$ remains in this class.

We use the following three notations for the asymptotic relations of arbitrary positive functions f and g: $f\left(x\right)=o\left(g\left(x\right)\right)$ means that $\underset{x\to \infty }{lim}f\left(x\right)/g\left(x\right)=0$; $f\left(x\right)\sim cg\left(x\right),c>0,$ means that $\underset{x\to \infty }{lim}f\left(x\right)/g\left(x\right)=c$; and $f\left(x\right)\asymp g\left(x\right)$ means that

$$0<\underset{x\to \infty }{lim\; inf}\frac{f\left(x\right)}{g\left(x\right)}⩽\underset{x\to \infty }{lim\; sup}\frac{f\left(x\right)}{g\left(x\right)}<\infty .$$

The rest of the paper is organized as follows. In Section 2, we desribe class of generalized subexponential distributions. Section 4 consists of some results on closure under randomly stopped sums for regularity classes related with generalized subexponential distributions. The main results of the paper are formulated in Section 3. The proofs of the main results are given in Section 5 and Section 6.

2. Generalized subexponentiality

Let $\xi$ be a r.v. defined on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with d.f. $F_{\xi }$.

•A d.f. $F_{\xi }$ of a real-valued r.v. is said to be generalized subexponential, denoted $F_{\xi }\in \mathcal{OS}$, if

$$\underset{x\to \infty }{lim\; sup}\frac{\overline{F_{\xi }\ast F_{\xi }}\left(x\right)}{{\overline{F}}_{\xi }\left(x\right)}<\infty ,$$

where $F_{\xi }\ast F_{\xi }$ denote the convolution of d.f. $F_{\xi }$ with it self, i.e.

$$F_{\xi }\ast F_{\xi }\left(x\right)=F_{\xi }^{\ast 2}\left(x\right):={\int }_{-\infty }^{\infty }{F}_{\xi }(x-y)\mathrm{d}{F}_{\xi }\left(y\right),x\in \mathbb{R}.$$

For distributions of non-negative r.v.s class $\mathcal{OS}$ was introduced by Klüppelberg [1] and later for real-valued r.v.s was studied by Shimura and Watanabe [2], Baltrūnas et al. [3], Watanabe and Yamamuro [4], Yu and Wang [5], Cheng and Wang [6], Lin and Wang [7], Konstantinides et al. [8], Mikutavičius and Šiaulys [9] among others.

In [2], the class of distributions $\mathcal{OS}$ is considered together with other distribution regularity classes. In that paper, several closedness properties of the class $\mathcal{OS}$ were proved. For example, it is shown that the class $\mathcal{OS}$ is not closed under convolution roots. This means that there exist r.v. $\xi$ such that n-fold convolution $F_{\xi }^{\ast n}\in \mathcal{OS}$ for all $n⩾2$, but $F_{\xi }\notin \mathcal{OS}$. In [3], the simple conditions are provided under which d.f. of the special form

$$F_{\xi }\left(x\right)=1-exp\left\{-\underset{0}{\overset{x}{\int }}q\left(u\right)\mathrm{d}u\right\}$$

belongs to the class $\mathcal{OS}$, where q is some integrable hazard rate function. For distributions of class $\mathcal{OS}$ the closure under tail-equivalence and the closure under convolution are established in [4]. The detailed proofs of these closures for non-negative r.v.s are presented in [1] and for real-valued r.v.s in [5]. The closure under convolution tail equivalence means that in case of independent r.v.s ${\xi }_1,{\xi }_2$ conditions $F_{{\xi }_1}\in \mathcal{OS}$, $F_{{\xi }_2}\in \mathcal{OS}$ imply that $F_{{\xi }_1}\ast F_{{\xi }_2}=F_{{\xi }_1+{\xi }_2}\in \mathcal{OS}$. The closure under tail-equivalence means that conditions $F_{{\xi }_1}\in \mathcal{OS}$, ${\overline{F}}_{{\xi }_1}\left(x\right)\asymp {\overline{F}}_{{\xi }_2}\left(x\right)$ imply $F_{{\xi }_2}\in \mathcal{OS}$.

A counterexample, showing that $F_{{\xi }_1},F_{{\xi }_2}\in \mathcal{OS}$ for independent r.v.s ${\xi }_1,{\xi }_2$ does not imply $F_{{\xi }_1\vee {\xi }_2}\in \mathcal{OS}$ can be found in [7]. Moreover in that paper, the closure under maximum is established which means that $F_{{\xi }_1},F_{{\xi }_2}\in \mathcal{OS}$ for independent r.v.s ${\xi }_1,{\xi }_2$ imply $F_{{\xi }_1\wedge {\xi }_2}\in \mathcal{OS}$. The authors of articles [8] and [9] consider when the distribution of the product of two independent random variables $\xi ,\theta$ belongs to the class $\mathcal{OS}$. For instance in [9], it is proved that d.f. $F_{\xi \theta }$ is generalized subexponential if $F_{\xi }\in \mathcal{OS}$ and $\theta$ is non-negative and not-degenerated at zero.

3. Main results

In this section, we formulate two theorems which are the main assertions of this paper. The first theorem deals to the case when the counting r.v. has a finite support.

Theorem 1.

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent r.v.s, and η be a counting r.v. independent of $\{{\xi }_1,{\xi }_2,\dots \}$. If η is bounded, $F_{{\xi }_1}\in \mathcal{OS}$, and for other indices $k\ne 1$ either $F_{{\xi }_k}\in \mathcal{OS}$ or ${\overline{F}}_{{\xi }_k}\left(x\right)=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right)$, then d.f. of randomly stopped sum $F_{S_{\eta }}$ belongs to the class $\mathcal{OS}$.

The case of unbounded support of counting r.v. is considered in the second theorem. In such a case, to be $F_{S_{\eta }}\in \mathcal{OS}$ we need the counting random variable to have a light tail.

Theorem 2.

Let $\{\eta ,{\xi }_1,{\xi }_2,...\}$ be independent random variables, where counting r.v. η be such that $\mathbb{E}{e}^{\lambda \eta }<\infty$ for all $\lambda >0$. Then $F_{S_{\eta }}\in OS$, if $F_{{\xi }_1}\in OS$ and one of the conditions below is satisfied:

$$\begin{array}{cc}\hfill & \left(\mathrm{i}\right)\mathbb{P}(\eta =1)>0\mathrm{and}\underset{x\to \infty }{lim\; sup}\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}<\infty ;\hfill \\ \hfill & \left(\mathrm{ii}\right)0<\underset{x\to \infty }{lim\; inf}\underset{k⩾1}{inf}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽\underset{x\to \infty }{lim\; sup}\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}<\infty .\hfill \end{array}$$

We will present the proofs of both theorems in Section 6. According to the statements of these theorems, many random variables with generalized subexponential distributions can be constructed. We will demonstrate such constructions in section ??

4. Similar results for related regularity classes

In this section, we will describe several classes of distributions related to the class $\mathcal{OS}$. For the described classes, we will present some results on their closure with respect to a randomly stopped sum. We note that for some classes, the closedness of the randomly stopped sum is studied only in the case where the summands are identically distributed.

The class of generalized subexponential distributions is the direct generalization of

$$\widehat{\mathcal{S}}=\bigcup _{\gamma ⩾0}\mathcal{S}\left(\gamma \right),$$

where $\mathcal{S}\left(0\right)=\mathcal{S}$ is the class of the subexponential distributions and $\left\{\mathcal{S}\right(\gamma )$, $\gamma >0\}$ are the convolution equivalent distributions classes.

•A d.f. $F_{\xi }$ of a non-negative r.v. ξ is said to be subexponential, denoted $F_{\xi }\in \mathcal{S}$, if

$$\overline{F_{\xi }\ast F_{\xi }}\left(x\right)\sim 2{\overline{F}}_{\xi }\left(x\right).$$

A d.f. $F_{\xi }$ of a real-valued r.v. ξ is called subexponential if the positive part of d.f.

$$F_{\xi }^{+}\left(x\right)=F_{\xi }\left(x\right){\mathbb{I}}_{[0,\infty )}\left(x\right)$$

belongs to the class $\mathcal{S}$.

The class of subexponential distributions was introduced by Chistyakov [10] and later considered by Athreya and Ney [11], Chover et al. [12,13], Embrechts and Goldie [14], Embrechts and Omey [15], Cline [16] and Cline and Samorodnitsky [17] among others.

•A d.f. $F_{\xi }$ of a real-valued r.v. ξ is said to be convolution equivalent with parameter $\gamma >0$, denoted $F_{\xi }\in \mathcal{S}\left(\gamma \right)$, if the following requirements are satisfyed

$$\begin{array}{cc}\hfill & \left(\mathrm{i}\right){\widehat{F}}_{\xi }\left(\gamma \right):={\int }_{-\infty }^{\infty }{\mathrm{e}}^{\gamma x}\mathrm{d}{F}_{\xi }\left(x\right)<\infty ,\hfill \\ \hfill & \left(\mathrm{ii}\right)\underset{x\to \infty }{lim}\frac{{\overline{F}}_{\xi }(x-y)}{{\overline{F}}_{\xi }\left(x\right)}={\mathrm{e}}^{\gamma y}\mathrm{for}\mathrm{all}y>0,\hfill \\ \hfill & \left(\mathrm{iii}\right)\underset{x\to \infty }{lim}\frac{\overline{F_{\xi }\ast F_{\xi }}\left(x\right)}{{\overline{F}}_{\xi }\left(x\right)}=2c_{\xi }\mathrm{for}\mathrm{some}\mathrm{constant}{c}_{\xi }\hfill \end{array}$$

The study of class $\mathcal{S}\left(\gamma \right)$ goes back to Chover et al. [12,13], Embrechts and Goldie [14], Klüppelberg [18]. It is well known that $F\in \mathcal{S}\left(\gamma \right)$ if and only if $F_{\xi }^{+}\in \mathcal{S}\left(\gamma \right)$, see Corollary 2.1(i) in [19], and the constant $c_{\xi }$ in the definition above is equal to ${\widehat{F}}_{\xi }\left(\gamma \right)$, see [19,20,21]. For $\gamma >0$ a standard example of d.f. in $\mathcal{S}\left(\gamma \right)$ is d.f. satisfying

$$\begin{array}{c}\hfill \widehat{F}\left(x\right)\underset{x\to \infty }{\sim }c{\mathrm{e}}^{-\gamma x}{x}^{-\alpha }\end{array}$$

with parameters $c>0,\gamma >0,\alpha >1$, see [22,23].

For the class $\mathcal{S}$ the following result is obtained in Theorem 3.37 of [24], see also [25,26,27,28].

Theorem 3.

Let $\{{\xi }_1,{\xi }_2,\cdots \}$ be a sequence of independent real-valued r.v.s with common distribution $F_{\xi }\in \mathcal{S}$, and let η be independent counting r.v. with expectation $\mathbb{E}\eta$, such that $\mathbb{E}{(1+\epsilon )}^{\eta }<\infty$ for some $\epsilon >0$. Then

$$\begin{array}{c}\hfill {\overline{F}}_{S_{\eta }}\left(x\right)\sim \mathbb{E}\eta {\overline{F}}_{\eta }\left(x\right),\end{array}$$

and $F_{S_{\eta }}\in \mathcal{S}$.

For the class $\mathcal{S}\left(\gamma \right)$ with $\gamma >0$ the following assertion is derived in Theorem C of [29], see also [30,31,32] for related results.

Theorem 4.

Let $\{{\xi }_1,{\xi }_2,\cdots \}$ independent real-valued r.v.s with common distribution $F_{\xi }\in \mathcal{S}\left(\gamma \right)$, $\gamma >0$, and let η be independent counting r.v. independent of $\{{\xi }_1,{\xi }_2,\cdots \}$. If

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\mathbb{P}(\eta =n)max\left\{{\left({\widehat{F}}_{\xi }\left(\gamma \right)+\epsilon \right)}^n,1\right\}<\infty \end{array}$$

for some $\epsilon >0$, then $F_{S_{\eta }}\in \mathcal{S}\left(\gamma \right)$.

We note that in the theorems 3 and 4 r.v.s in the sequences $\{{\xi }_1,{\xi }_2,\cdots \}$ are identically distributed. However, there are related regularity classes for which similar results can be obtained in cases where r.v.s in $\{{\xi }_1,{\xi }_2,\cdots \}$ are not necessarily identically distributed. Here we discuss two such classes.

•A d.f. $F_{\xi }$ of a real valued r.v. ξ is said to be dominatedly varying, denoted $F_{\xi }\in \mathcal{D}$, if

$$\underset{x\to \infty }{lim\; sup}\frac{\overline{F_{\xi }}\left(yx\right)}{{\overline{F}}_{\xi }\left(x\right)}<\infty$$

for all (or, equivalently, for some) $y\in (0,1)$.

•A d.f. $F_{\xi }$ of a real valued r.v. ξ is said to be exponential-like-tailed, denoted $F_{\xi }\in \mathcal{L}\left(\gamma \right)$, if

$$\underset{x\to \infty }{lim}\frac{\overline{F_{\xi }}(x-y)}{{\overline{F}}_{\xi }\left(x\right)}={\mathrm{e}}^{\gamma y}$$

for all $y>0$.

Class of dominatedly varying d.f.s $\mathcal{D}$ was introduced by Feller [33] and later considered in [4,34,35,36,37,38,39] among others. The class of long-tailed d.f.s $\mathcal{L}\left(0\right)$ was introduced by Chistyakov [10] in the context of branching processes. The class $\mathcal{L}\left(\gamma \right)$ with $\gamma >0$ was introduced by Chover et al. [12,13]. Later the various properties of long-tailed and exponential-like-tailed d.f.s were considered in [1,19,24,29,38,40,41] for instance. Here we recall only that $\mathcal{L}\left(0\right)\cap \mathcal{D}\subset \mathcal{S}$ and $\mathcal{S}\left(\gamma \right)\subset \mathcal{L}\left(\gamma \right)$.

The following assertion on $F_{S_{\eta }}\in \mathcal{D}$ is presented in Theorem 4 of [42].

Theorem 5.

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent real-valued r.v.s with common d.f. $F_{\xi }\in \mathcal{D}$, and let η be a counting r.v. independent of $\{{\xi }_1,{\xi }_2,\dots \}$. Then $F_{S_{\eta }}\in \mathcal{D}$ if $\mathbb{E}{\eta }^{p+1}<\infty$ for some

$$p>J_{F_{\xi }}^{+}:=-\underset{y\to \infty }{lim}\frac{1}{logy}log\underset{x\to \infty }{lim\; inf}\frac{{\overline{F}}_{\xi }\left(xy\right)}{{\overline{F}}_{\xi }\left(x\right)}.$$

In the inhomogeneous case, when sumands are not necessary identically distributed, the following statement is obtained in Theorem 2.1 of [43].

Theorem 6.

Let $\{{\xi }_1,{\xi }_2,\cdots \}$ be a sequence independent nonnegative r.v.s, and let η be a counting r.v. independent of $\{{\xi }_1,{\xi }_2,\cdots \}$. Then $F_{S_{\eta }}\in \mathcal{D}$ if the following three conditions are satisfied:

(i) $F_{{\xi }_{\varkappa }}\in \mathcal{D}$ for some $\varkappa \in \mathrm{supp}\left(\eta \right):=\{n\in {\mathbb{N}}_0:\mathbb{P}(\eta =n)>0\}$,

(ii)$\underset{x\to \infty }{lim\; sup}\underset{n>\varkappa }{sup}\frac{1}{n{\overline{F}}_{{\xi }_{\varkappa }}\left(x\right)}\sum _{i=1}^n{\overline{F}}_{{\xi }_i}\left(x\right)<\infty$,

(iii)$\mathbb{E}{\eta }^{p+1}<\infty$ for some $p>J_{F_{{\xi }_{\varkappa }}}^{+}$.

Examples of conditions for the function $F_{S_{\eta }}$ to belong to the class $\mathcal{L}\left(\gamma \right)$ are given in the theorems below. Theorem 7 proved in [42] present conditions for the homogeneous case for class $\mathcal{L}=\mathcal{L}\left(0\right)$, while Theorem 8 proved in [44] gives conditions for the inhomogeneous case for class $\mathcal{L}\left(\gamma \right)$ with $\gamma ⩾0$.

Theorem 7.

Suppose that $\{{\xi }_1,{\xi }_2,\dots \}$ are independent nonnegative r.v.s with common distribution $F_{\xi }\in \mathcal{L}$, and let η be a counting r.v. independent of $\{{\xi }_1,{\xi }_2,\dots \}$. If

$$\begin{array}{c}\hfill {\overline{F}}_{\eta }\left(\delta x\right)=o\left(\sqrt{x}{\overline{F}}_{\xi }\left(x\right)\right)\end{array}$$

for any $\delta \in (0,1)$, then $F_{S_{\eta }}\in \mathcal{L}$.

Theorem 8.

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent r.v.s such that for some $\gamma ⩾0$

$$\underset{k⩾1}{sup}|\frac{{\overline{F}}_{{\xi }_k}(x+y)}{{\overline{F}}_{{\xi }_k}\left(x\right)}-{\mathrm{e}}^{-\gamma y}|\underset{x\to \infty }{\to }0$$

for each fixed $y>0$, and let η be a counting r.v. independent of $\{{\xi }_1,{\xi }_2,\dots \}$. If

$$\begin{array}{c}\hfill \frac{\mathbb{P}(\eta =k+1)}{\mathbb{P}(\eta =k)}\underset{k\to \infty }{\to }0,\end{array}$$

then $F_{S_{\eta }}\in \mathcal{L}\left(\gamma \right)$.

In the context of the randomly stopped sums the class $OS$ was considered by Shimura and Watanabe [2]. In Proposition 3.1 of that paper the following assertion is presented.

Theorem 9.

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of nonnegative independent r.v.s with common d.f. $F_{\xi }$, and let η be a counting r.v. such that

$$\begin{array}{c}\hfill \mathbb{P}(\eta >1)>0,sup\left\{x⩾1:\sum _{k=0}^{\infty }\mathbb{P}(\eta =k)x^k<\infty \right\}=\infty .\end{array}$$

Then $F_{\xi }\in \mathcal{OS}$ if and only if ${\overline{F}}_{S_{\eta }}\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_{\xi }\left(x\right)$.

From the information presented, it can be seen that our main theorems 1 and 2 in fact are inhomogeneous versions of the formulated theorem 9.

5. Auxiliary lemmas

In this section, we will present and prove some auxiliary lemmas that will be applied to the derivations of the main theorems 1 and 2.

Lemma 1.

Let X and Y be two real valued r.v.s with corresponding d.f.s $F_X$ and $F_Y$. The following statements hold:

(i) $F_X\in \mathcal{OS}$ if and only if $\underset{x\in \mathbb{R}}{sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}<\infty .$

(ii) If $F_X\in \mathcal{OS}$ and ${\overline{F}}_Y\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_X\left(x\right)$, then $F_Y\in \mathcal{OS}$.

(iii) If $F_X\in \mathcal{OS}$ and $F_Y\in \mathcal{OS}$, then $F_X\ast F_Y\in \mathcal{OS}$.

(iv) If $F_X\in \mathcal{OS}$, then $F_X\in \mathcal{OL}$ i.e. $\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_X(x-1)}{{\overline{F}}_X\left(x\right)}=1$.

(v) If $F_X\in \mathcal{OS}$ and ${\overline{F}}_Y\left(x\right)=O\left({\overline{F}}_X\left(x\right)\right)$, then $F_X\ast F_Y\in \mathcal{OS}$ and $\overline{F_X\ast F_Y}\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_X\left(x\right)$.

Proof. A large part of the properties of the class $\mathcal{OS}$ listed in Lemma 1 can be found, for instance, in [1,2,4,5]. However, for the sake of exposition completeness, we present the full proof of the formulated lemma.

Part (i). If $F_X\in OS$, then

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim\; sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}<\infty \end{array}$$

(1)

according to definition. This estimate implies that ${\overline{F}}_X\left(x\right)>0$ for each $x\in \mathbb{R}$. In addition, the inequality (1) gives that

$$\begin{array}{c}\hfill \frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽M\end{array}$$

if $x⩾x_M$ for some M and $x_M$.

If $x<x_M$, then, obviously, ${\overline{F}}_X\left(x\right)⩾{\overline{F}}_X\left(x_M\right)$ and $\overline{F_X\ast F_X}\left(x\right)⩽1$.

Therefore, for each $x\in \mathbb{R}$ we get that

$$\begin{array}{c}\hfill \frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽max\left\{M,\frac{1}{{\overline{F}}_X\left(x_M\right)}\right\}<\infty \end{array}$$

because ${\overline{F}}_X\left(x_M\right)>0$. The last estimate finishes the proof of the part (i) because the condition

$$\begin{array}{c}\hfill \underset{x\in \mathbb{R}}{sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}<\infty \end{array}$$

implies (1) obviously.

Part (ii). The condition ${\overline{F}}_Y\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_X\left(x\right)$ implies

$$\underset{x\to \infty }{lim\; inf}\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}>0\mathrm{and}\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}<\infty .$$

(2)

It follows from this that

$$\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽M,x⩾x_M,$$

for some M and $x_M$. If $x<x_M$, then

$$\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽\frac{1}{{\overline{F}}_X\left(x_M\right)}<\infty$$

because $F_X\in \mathcal{OS}$. According to the derived estimates

$$\underset{x\in \mathbb{R}}{sup}\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}=max\left\{M,\frac{1}{{\overline{F}}_X\left(x_M\right)}\right\}=C<\infty .$$

Therefore for each $x\in \mathbb{R}$

$$\begin{array}{cc}\hfill \overline{F_Y\ast F_Y}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}\frac{{\overline{F}}_Y(x-y)}{{\overline{F}}_X(x-y)}{\overline{F}}_X(x-y)\mathrm{d}{F}_Y\left(y\right)⩽C\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_X(x-y)\mathrm{d}{F}_Y\left(y\right)\hfill \\ \hfill & =C\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_Y(x-y)\mathrm{d}{F}_X\left(y\right)=C\underset{-\infty }{\overset{\infty }{\int }}\frac{{\overline{F}}_Y(x-y)}{{\overline{F}}_X(x-y)}{\overline{F}}_X(x-y)\mathrm{d}{F}_X\left(y\right)\hfill \\ \hfill & ⩽C^2\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_X(x-y)\mathrm{d}{F}_X\left(y\right)=C^2\overline{F_X\ast F_X}\left(x\right).\hfill \end{array}$$

This estimate implies that

$$\begin{array}{cc}\hfill \underset{x\to \infty }{lim\; sup}\frac{\overline{F_Y\ast F_Y}\left(x\right)}{{\overline{F}}_Y\left(x\right)}& ⩽C^2\underset{x\to \infty }{lim\; sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_Y\left(x\right)}\hfill \\ & ⩽C^2\underset{x\to \infty }{lim\; sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}\frac{1}{\underset{x\to \infty }{lim\; inf}\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}}<\infty \hfill \end{array}$$

due to the assumption $F_X\in \mathcal{OS}$ and the first inequality in (2). The last estimate gives that d.f. $F_Y$ belongs to the class $\mathcal{OS}$. Part (ii) of the lemma is proved.

Part (iii). According to part (i) we have that

$$\underset{x\in \mathbb{R}}{sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}=C_1<\infty \mathrm{and}\underset{x\in \mathbb{R}}{sup}\frac{\overline{F_Y\ast F_Y}\left(x\right)}{{\overline{F}}_Y\left(x\right)}=C_2<\infty$$

Let $X_1,X_2,Y_1,Y_2$ be independent r.v.s. Suppose that $X_1$, $X_2$ are distributed according to the d.f. $F_X$, and $Y_1$, $Y_2$ are distributed according to the d.f. $F_Y$. For each $x\in \mathbb{R}$ we get

$$\begin{array}{cc}\hfill \overline{{\left({(F_X\ast F_Y)}^{\ast }\right)}^2}\left(x\right)& =\overline{(F_X\ast F_Y)\ast (F_X\ast F_Y)}\left(x\right)=\mathbb{P}(X_1+Y_1+X_2+Y_2>x)\hfill \\ \hfill & =\mathbb{P}(X_1+X_2+Y_1+Y_2)>x)=\underset{-\infty }{\overset{\infty }{\int }}\mathbb{P}(X_1+X_2>x-y)\mathrm{d}\mathbb{P}(Y_1+Y_2⩽y)\hfill \\ \hfill & =\underset{-\infty }{\overset{\infty }{\int }}\frac{\overline{F_X\ast F_X}(x-y)}{{\overline{F}}_X(x-y)}{\overline{F}}_X(x-y)\mathrm{d}\mathbb{P}(Y_1+Y_2⩽y)\hfill \\ \hfill & ⩽C_1\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_X(x-y)\mathrm{d}\mathbb{P}(Y_1+Y_2⩽y)=C_1\mathbb{P}(X_1+Y_1+Y_2>x)\hfill \\ & =C_1\underset{-\infty }{\overset{\infty }{\int }}\frac{\overline{F_Y\ast F_Y}(x-y)}{{\overline{F}}_Y(x-y)}{\overline{F}}_Y(x-y)\mathrm{d}\mathbb{P}(X_1⩽y)\hfill \\ \hfill & ⩽C_1{C}_2\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_Y(x-y)d{F}_X\left(y\right)=C_1{C}_2\overline{F_X\ast F_Y}\left(x\right).\hfill \end{array}$$

Hence

$$\underset{x\in \mathbb{R}}{sup}\frac{\overline{{\left({(F_X\ast F_Y)}^{\ast }\right)}^2}\left(x\right)}{\overline{F_X\ast F_Y}\left(x\right)}⩽C_1{C}_2$$

implying that $F_X\ast F_Y\in \mathcal{OS}$ by part (i). Part (iii) of the lemma is proved.

Part (iv). Due to the part (i)

$$\underset{x\in \mathbb{R}}{sup}\frac{\overline{F_X\ast F_X}\left(x\right)}{\overline{F_X}\left(x\right)}=C_3<\infty .$$

In addition, for $x>2$, we obtain

$$\begin{array}{cc}\hfill \overline{F_X\ast F_X}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_X(x-t)\mathrm{d}{F}_X\left(t\right)⩾\underset{(1,x]}{\int }{\overline{F}}_X(x-t)\mathrm{d}{F}_X\left(t\right)\hfill \\ & ⩾{\overline{F}}_X(x-1)(F_X\left(x\right)-F_X\left(1\right))\hfill \end{array}$$

When x is large enough we have $F\left(x\right)-F\left(1\right)>0$, and, therefore,

$$\frac{{\overline{F}}_X(x-1)}{{\overline{F}}_X\left(x\right)}⩽\frac{\overline{F_X\ast F_X}\left(x\right)}{{\overline{F}}_X\left(x\right)}\frac{1}{F_X\left(x\right)-F_X\left(1\right)}.$$

Hence

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_X(x-1)}{{\overline{F}}_X\left(x\right)}⩽\frac{C_3}{{\overline{F}}_X\left(1\right)}<\infty ,\end{array}$$

and part (iv) of the lemma is proved.

Part (v). Since ${\overline{F}}_Y\left(x\right)=O\left({\overline{F}}_X\left(x\right)\right)$, we have

$$\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽M,x>x_M,$$

with certain constants M and $x_M$. If $x⩽x_M$, then

$$\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽\frac{1}{{\overline{F}}_X\left(x_M\right)}<\infty$$

because $F_X\in \mathcal{OS}$ implies ${\overline{F}}_X\left(x_M\right)>0$. From the both above inequalities it follows that

$$\underset{x\in \mathbb{R}}{sup}\frac{{\overline{F}}_Y\left(x\right)}{{\overline{F}}_X\left(x\right)}⩽max\left\{M,\frac{1}{{\overline{F}}_X\left(x_M\right)}\right\}=C_4$$

Consequently, for $x\in \mathbb{R}$ we get

$$\begin{array}{cc}\hfill \overline{F_X\ast F_Y}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_Y(x-y)\mathrm{d}{F}_X\left(y\right)⩽C_4\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_X(x-y)\mathrm{d}{F}_X\left(y\right)\hfill \\ \hfill & =C_4\overline{F_X\ast F_X}\left(x\right)⩽C_5\overline{F_X}\left(x\right)\hfill \end{array}$$

(3)

with some positive constant $C_5$, where the last step in the above derivation follows from part (i) of the lemma.

On the other hand, there exists a real $b\in \mathbb{R}$ for which

$${\overline{F}}_Y\left(b\right)=1-F_Y\left(b\right)⩾\frac{1}{2}$$

For this b, we get

$$\begin{array}{cc}\hfill \overline{F_X\ast F_Y}\left(x\right)& ⩾\underset{(b,\infty )}{\int }{\overline{F}}_X(x-y)\mathrm{d}{F}_Y\left(y\right)⩾{\overline{F}}_X(x-b)\underset{(b,\infty )}{\int }\mathrm{d}{F}_Y\left(y\right)\hfill \\ \hfill & ={\overline{F}}_X(x-b){\overline{F}}_Y\left(b\right)⩾\frac{1}{2}{\overline{F}}_X\left(x\right)\frac{{\overline{F}}_X(x-b)}{{\overline{F}}_X\left(x\right)}\hfill \end{array}$$

Hence,

$$\underset{x\to \infty }{lim\; inf}\frac{\overline{F_X\ast F_Y}\left(x\right)}{{\overline{F}}_X\left(x\right)}⩾\frac{1}{2}\underset{x\to \infty }{lim\; inf}\frac{{\overline{F}}_X(x-b)}{{\overline{F}}_X\left(x\right)}.$$

(4)

In part (iv) of the lemma we proved that $F_X\in \mathcal{OL}$. It is easy to verify that

$$\begin{array}{c}\hfill F_X\in \mathcal{OL}\iff \underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_X(x-1)}{{\overline{F}}_X\left(x\right)}<\infty {\overline{F}}_X(x-b)\underset{x\to \infty }{\asymp }{\overline{F}}_X\left(x\right)\mathrm{for}\mathrm{each}b\in \mathbb{R}.\end{array}$$

Therefore, the estimate (4) implies that

$$\begin{array}{c}\hfill \underset{x\to \infty }{lim\; inf}\frac{\overline{F_X\ast F_Y}\left(x\right)}{{\overline{F}}_X\left(x\right)}>0.\end{array}$$

(5)

From (3) and (5) inequalities it follows that $\overline{F_X\ast F_Y}\left(x\right)\underset{x\to \infty }{\asymp }\overline{F_X}\left(x\right).$ Moreover by part (ii) of the lemma $F_X\ast F_Y\in \mathcal{OS}$. This finish the proof of the last part of the lemma. □

Lemma 2.

Let $\{{\xi }_1,{\xi }_2...\}$ be a sequence of independent r.v.s, for which $F_{{\xi }_1}\in \mathcal{OS}$, and for others indices $k⩾2$ either $F_{{\xi }_k}\in \mathcal{OS}$ or ${\overline{F}}_{{\xi }_k}=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right)$. Then $F_{S_n}\in \mathcal{OS}$ for all $n\in \mathbb{N}$.

Proof. If $n=1$,then the statement is obvious because $S_1={\xi }_1$. If $n=2$, then two options are possible $F_{{\xi }_2}\in \mathcal{OS}$ or ${\overline{F}}_{{\xi }_2}=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right)$. In the first case $F_{S_2}=F_{{\xi }_1}\ast F_{{\xi }_2}\in \mathcal{OS}$ according to the part (iii) of Lemma 1. In the second case $F_{S_2}\in \mathcal{OS}$ by the part (v) of the same lemma.

Let now $n>2$. Denote

$$\mathcal{K}=\{k\in \{2,...,n\}:{\overline{F}}_{{\xi }_k}\left(x\right)=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right)\}.$$

Initially assume that the set $\mathcal{K}$ is empty. In such a case, $F_{{\xi }_k}\in \mathcal{OS}$ for all indices $k\in {\mathcal{K}}^c=\{1,2,3,\dots ,n\}$. By the part (iii) of Lemma 1 we get that $F_{S_n}\in \mathcal{OS}$.

Let now the index set $\mathcal{K}=\{k_1,k_2,...k_r\}\subset \{1,...n\}$ is not empty. Since

$${\overline{F}}_{{\xi }_{k_1}}\left(x\right)=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right),$$

part (v) of Lemma 1 implies that

$$F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\in \mathcal{OS},$$

(6)

and

$$\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}}\left(x\right)\asymp {\overline{F}}_{{\xi }_1}\left(x\right).$$

(7)

According the relation (7)

$$\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_{{\xi }_{k_2}}\left(x\right)}{\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}}\left(x\right)}⩽\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_{{\xi }_{k_2}}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}\frac{1}{\underset{x\to \infty }{lim\; inf}\frac{\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}}<\infty$$

because ${\overline{F}}_{{\xi }_{k_2}}\left(x\right)=O\left({\overline{F}}_{{\xi }_1}\left(x\right)\right)$. This means that

$${\overline{F}}_{{\xi }_{k_2}}\left(x\right)=O\left(\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}}\left(x\right)\right).$$

Hence according to (6) and part (v) of Lemma 1 we get

$$F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\ast F_{{\xi }_{k_2}}=(F_{{\xi }_1}\ast F_{{\xi }_{k_1}})\ast F_{{\xi }_{k_2}}\in \mathcal{OS},$$

and

$$\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\ast F_{{\xi }_{k_2}}}\left(x\right)\asymp \overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}}\left(x\right).$$

Continuing the process we obtain

$$F_{\mathcal{K}}:=F_{{\xi }_1}\ast \prod _{j=1}^r\ast F_{{\xi }_{k_j}}=F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\ast F_{{\xi }_{k_2}}\ast ...\ast F_{{\xi }_{k_r}}\in \mathcal{OS},$$

and

$$\overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\ast F_{{\xi }_{k_2}}\ast ...\ast F_{{\xi }_{k_r}}}\left(x\right)\asymp \overline{F_{{\xi }_1}\ast F_{{\xi }_{k_1}}\ast F_{{\xi }_{k_2}}\ast ...\ast F_{{\xi }_{k_r-1}}}\left(x\right).$$

For the remaining indices $k\in {\mathcal{K}}^c=\{2,3,...,n\}\setminus \{k_1,k_2,...,k_r\}$ d.f. $F_{{\xi }_k}\in \mathcal{OS}$. By the part (iii) of Lemma 1 we get

$$F_{{\mathcal{K}}^c}:=\prod _{k\in {\mathcal{K}}^c}\ast F_{{\xi }_k}\in \mathcal{OS}.$$

Using part (iii) of Lemma 1 again we derive that

$$F_{S_n}=F_{\mathcal{K}}\ast F_{{\mathcal{K}}^c}\in \mathcal{OS}.$$

This finish the proof of Lemma 2. □

Lemma 3.

Let ${\xi }_1,{\xi }_2,...$ be a sequence of independent random variables,for which $F_{{\xi }_1}\in OS$ and

$$\underset{x\to \infty }{lim\; sup}\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}<\infty$$

(8)

Then there exists a constant $\widehat{C}$ for which

$${\overline{F}}_{S_n}\left(x\right)⩽{\widehat{C}}^{n-1}{\overline{F}}_{{\xi }_1}\left(x\right)$$

(9)

for all $x\in \mathbb{R}$ and for all $n⩾2$.

Proof. The condition (8) implies that

$$\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽C_6$$

for all $x⩾A$ with some positive constants $C_6$ and A. If $x<A$,then

$$\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽\frac{1}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽\frac{1}{{\overline{F}}_{{\xi }_1}\left(A\right)}<\infty .$$

Therefore, for each $x\in \mathbb{R}$

$$\underset{k⩾1}{sup}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽max\left\{C_6,\frac{1}{{\overline{F}}_{{\xi }_1}\left(A\right)}\right\}:=C_7.$$

(10)

In addition, the part (i) of Lemma 1 gives that

$$\begin{array}{c}\hfill \overline{F_{{\xi }_1}\ast F_{{\xi }_1}}\left(x\right)⩽C_8{\overline{F}}_{{\xi }_1}\left(x\right)\end{array}$$

(11)

for all $x\in \mathbb{R}$ with some positive constant $C_8$.

We will prove the inequality (9) with constant $\widehat{C}=C_7{C}_8$. If $n=1$, the inequality (9) holds evidently because ${\overline{F}}_{S_1}\left(x\right)={\overline{F}}_{{\xi }_1}\left(x\right).$ If $n=2$, then by (10) and (11) for $x\in \mathbb{R}$ we have

$$\begin{array}{cc}\hfill {\overline{F}}_{S_2}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{{\xi }_2}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)⩽C_7\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{{\xi }_1}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)\hfill \\ \hfill & =C_7\overline{F_{{\xi }_1}\ast F_{{\xi }_1}}\left(x\right)⩽\widehat{C}{\overline{F}}_{{\xi }_1}\left(x\right).\hfill \end{array}$$

Suppose now that the inequality (9) holds for $n=m⩾2$, i.e.

$$\frac{{\overline{F}}_{S_m}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽{\widehat{C}}^{m-1},x\in \mathbb{R}.$$

After choosing $n=m+1$, from this assumption and from (10), (11) we get

$$\begin{array}{cc}\hfill {\overline{F}}_{S_{m+1}}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{S_m}(x-y)\mathrm{d}{F}_{{\xi }_m+1}\left(y\right)⩽{\widehat{C}}^{m-1}\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{{\xi }_1}(x-y)\mathrm{d}{F}_{{\xi }_m+1}\left(y\right)\hfill \\ & ={\widehat{C}}^{m-1}\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{{\xi }_m+1}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)⩽{\widehat{C}}^{m-1}{C}_7\underset{-\infty }{\overset{\infty }{\int }}{\overline{F}}_{{\xi }_1}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)\hfill \\ & ={\widehat{C}}^{m-1}{C}_7\overline{F_{{\xi }_1}\ast F_{{\xi }_1}}\left(x\right)⩽{\widehat{C}}^m{\overline{F}}_{{\xi }_1}\left(x\right),x\in \mathbb{R}.\hfill \end{array}$$

According to the induction principle, the inequality (9) holds for all $n\in \mathbb{N}$. Lemma 3 is proved. □

6. Proofs of the main results

In this section, we present proofs of the main results op the paper.

Proof of Theorem 1. suppose that $\mathbb{P}(\eta \in \{0,1,\dots ,L\left\}\right)=1$ for some $L\in \mathbb{N}$. We have

$${\overline{F}}_{S_{\eta }}\left(x\right)=\sum _{n=1}^L\mathbb{P}(\eta =n){\overline{F}}_{S_n}\left(x\right),x\in \mathbb{R}.$$

Let

$$L^{\ast }=max\{k\in \{1,2,...,L\}:\mathbb{P}(\eta =k)>0\}.$$

For each $x\in \mathbb{R}$

$$\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{S_{L^{\ast }}}\left(x\right)}⩾\frac{\mathbb{P}(\eta =L^{\ast }){\overline{F}}_{S_{L^{\ast }}}\left(x\right)}{{\overline{F}}_{S_{L^{\ast }}}\left(x\right)}=\mathbb{P}(\eta =L^{\ast })>0.$$

(12)

On the other hand

$${\overline{F}}_{S_{\eta }}\left(x\right)=\sum _{k=0}^{L^{\ast }-1}\mathbb{P}(\eta =L^{\ast }-k)\mathbb{P}(S_{L^{\ast }-k}>x)$$

(13)

For any random variable ${\xi }_k$, $k\in \{1,2,\dots ,L^{\ast }\}$, there exists a negative number $-a_k$, for which $\mathbb{P}({\xi }_k⩾-a_k)⩾1/2$. We have

$$\begin{array}{cc}\hfill \mathbb{P}(S_{L^{\ast }-1}>x)& =\mathbb{P}(S_{L^{\ast }-1}-a_{L^{\ast }}>x-a_{L^{\ast }},{\xi }_{L^{\ast }}⩾-a_{L^{\ast }})+\mathbb{P}(S_{L^{\ast }-1}>x,{\xi }_{L^{\ast }}<-a_{L^{\ast }})\hfill \\ \hfill & ⩽\mathbb{P}(S_{L^{\ast }}>x-a_{L^{\ast }})+\mathbb{P}(S_{L^{\ast }-1}>x)\mathbb{P}({\xi }_{L^{\ast }}<-a_{L^{\ast }})\hfill \end{array}$$

From this we derive that

$$\mathbb{P}(S_{L^{\ast }-1}>x)⩽2\mathbb{P}(S_{L^{\ast }}>x-a_{L^{\ast }})$$

for each $x\in \mathbb{R}$. Similarly,

$$\mathbb{P}(S_{L^{\ast }-2}>x)⩽2\mathbb{P}(S_{L^{\ast }-1}>x-a_{L^{\ast }-1})⩽4\mathbb{P}(S_{L^{\ast }}>x-a_{L^{\ast }-1}-a_{L^{\ast }})$$

also for each real number x. Continuing the process we obtain

$$\mathbb{P}(S_{L^{\ast }-k}>x)⩽2^k\mathbb{P}\left(S_{L^{\ast }}>x-\sum _{j=0}^{k-1}{a}_{L^{\ast }-j}\right)$$

for all $x\in \mathbb{R}$ and for all $k=1,2,...,L^{\ast }-1$. After inserting the obtained estimates into inequality (13), we get that

$$\begin{array}{cc}\hfill {\overline{F}}_{S_{\eta }}\left(x\right)& ⩽\sum _{k=0}^{l^{\ast }-1}\mathbb{P}(\eta =L^{\ast }-k)2^k\mathbb{P}(S_{L^{\ast }}>x-\sum _{j=0}^{k-1}{a}_{L^{\ast }-j})\hfill \\ & ⩽\mathbb{P}(S_{L^{\ast }}>x-a^{★})\sum _{k=0}^{L^{\ast }-1}{2}^k\mathbb{P}(\eta =L^{\ast }-k)\hfill \\ \hfill & =C^{\ast }{\overline{F}}_{S_{L^{\ast }}}(x-a^{\ast }),\hfill \end{array}$$

where

$$C^{\ast }=\sum _{k=0}^{L^{\ast }-1}{2}^k\mathbb{P}(\eta =L^{\ast }-k),\mathrm{and}{a}^{\ast }=\sum _{j=2}^{L^{\ast }}{a}_j.$$

Consequently, for all x

$$\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{S_{L^{\ast }}}\left(x\right)}⩽\frac{C^{\ast }{\overline{F}}_{S_{L^{\ast }}}(x-a^{★})}{{\overline{F}}_{S_{L^{\ast }}}\left(x\right)}$$

(14)

By Lemma 2 and part (iv) of Lemma 1 we have that $F_{S_{L^{\ast }}}\in OS\subset OL$. Therefore

$$\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{S_{L^{\ast }}}\left(x\right)}<\infty$$

By (12) and (14) we have,that

$${\overline{F}}_{S_{\eta }}\left(x\right)\asymp {\overline{F}}_{S_{L^{\ast }}}\left(x\right)$$

Therefore $F_{S_{\eta }}\in OS$ together with $F_{S_{L^{\ast }}}$ by the part (ii) of Lemma 1. Theorem 1 is proved. □

Proof of Theorem 2. Part (i) Whereas

$${\overline{F}}_{S_{\eta }}\left(x\right)=\sum _{n=1}^{\infty }\mathbb{P}(\eta =n){\overline{F}}_{S_n}\left(x\right),$$

by Lemma 3 for all real numbers x we obtain

$$\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽\frac{\sum _{n=1}^{\infty }{\widehat{C}}^{n-1}\mathbb{P}(\eta =n){\overline{F}}_{{\xi }_1}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩽\mathbb{E}{\mathrm{e}}^{\widehat{C}\eta }<\infty ,$$

(15)

where $\widehat{C}$ is some positive constant.

On the other hand

$${\overline{F}}_{S_{\eta }}\left(x\right)⩾\mathbb{P}(\eta =1){\overline{F}}_{{\xi }_1}\left(x\right).$$

Hence under conditions of part (i), we have that ${\overline{F}}_{S_{\eta }}\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_{{\xi }_1}\left(x\right)$. Therefore $F_{S_{\eta }}\in OS$ according to part (ii) of Lemma 1. Part (i) of Theorem 2 is proved.

Part(ii). If $\mathbb{P}(\eta =1)>0$, then assertion of this part follows from the proved part (i). Since $\mathbb{E}{\mathrm{e}}^{\lambda \eta }<\infty$ for each $\lambda >0$, the inequality (15) implies that

$$\underset{x\to \infty }{lim\; sup}\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}<\infty .$$

(16)

In addition, conditions of part (ii) of the theorem give that

$$\underset{k⩾1}{inf}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩾\Delta$$

for all $x⩾x_{\Delta }$ and some positive $\Delta$. If $x<x_{\Delta }$, then

$$\underset{k⩾1}{inf}\frac{{\overline{F}}_{{\xi }_k}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}⩾\underset{k⩾1}{inf}\frac{{\overline{F}}_{{\xi }_k}\left(x_{\Delta }\right)}{{\overline{F}}_{{\xi }_1}\left(x_{\Delta }\right)}{\overline{F}}_{{\xi }_1}\left(x_{\Delta }\right)⩾\Delta {\overline{F}}_{{\xi }_1}\left(x_{\Delta }\right):=\tilde{C}>0$$

due to the assumption $F_{{\xi }_1}\in \mathcal{OS}$. The derived inequalities imply that

$${\overline{F}}_{{\xi }_k}\left(x\right)⩾\tilde{C}{\overline{F}}_{{\xi }_1}\left(x\right)$$

for some positive constant $\tilde{C}$, and for all $x\in \mathbb{R}$, $k\in \{1,2,\dots \}$.

Using the last estimate we get

$$\begin{array}{cc}\hfill {\overline{F}}_{S_2}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}\frac{{\overline{F}}_{{\xi }_2}(x-y)}{{\overline{F}}_{{\xi }_1}(x-y)}{\overline{F}}_{{\xi }_1}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)\hfill \\ \hfill & ⩾\tilde{C}\overline{F_{{\xi }_1}\ast F_{{\xi }_1}}\left(x\right)⩾\tilde{C}{\overline{F}}_{{\xi }_1}\left(0\right){\overline{F}}_{{\xi }_1}\left(x\right),x\in \mathbb{R}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\overline{F}}_{S_3}\left(x\right)& =\underset{-\infty }{\overset{\infty }{\int }}\frac{{\overline{F}}_{S_2}(x-y)}{{\overline{F}}_{{\xi }_1}(x-y)}{\overline{F}}_{{\xi }_1}(x-y)\mathrm{d}{F}_{{\xi }_1}\left(y\right)\hfill \\ \hfill & ⩾\tilde{C}{\overline{F}}_{{\xi }_1}\left(0\right)\overline{F_{{\xi }_1}\ast F_{{\xi }_1}}\left(x\right)⩾\tilde{C}{\left({\overline{F}}_{{\xi }_1}\left(0\right)\right)}^2{\overline{F}}_{{\xi }_1}\left(x\right),x\in \mathbb{R}.\hfill \end{array}$$

Continuing process we obtain

$${\overline{F}}_{S_n}\left(x\right)⩾\tilde{C}{\left({\overline{F}}_{{\xi }_1}\left(0\right)\right)}^{n-1}{\overline{F}}_{{\xi }_1}\left(x\right)$$

for all $x\in \mathbb{R}$ and $n\in \{2,3,\dots \}$.

Therefore,

$$\begin{array}{cc}\hfill \underset{x\to \infty }{lim\; inf}\frac{{\overline{F}}_{S_{\eta }}\left(x\right)}{{\overline{F}}_{{\xi }_1}\left(x\right)}& ⩾\underset{x\to \infty }{lim\; inf}\frac{\mathbb{P}(\eta =\tilde{L}){\overline{F}}_{S_{\tilde{L}}}}{{\overline{F}}_{{\xi }_1}\left(x\right)}\hfill \\ \hfill & ⩾\mathbb{P}(\eta =\tilde{L})\tilde{C}{\left({\overline{F}}_{{\xi }_1}\left(0\right)\right)}^{\tilde{L}-1}>0,\hfill \end{array}$$

(17)

where $\tilde{L}=min\{n⩾2:\mathbb{P}(\eta =n)>0\}$.

The derived inequalities (16) and (17) imply ${\overline{F}}_{S_{\eta }}\left(x\right)\underset{x\to \infty }{\asymp }{\overline{F}}_{{\xi }_1}\left(x\right)$. By part (ii) of Lemma 1 we get $F_{S_{\eta }}\in \mathcal{OS}$. Theorem 2 is proved.