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 η be a counting r.v., that is, an integer-valued, nonnegative, and nondegenerate at zero r.v. In addition, suppose that the r.v. η and r.v.s $\{\xi _{1},\xi _{2},\dots \}$ are independent. Let $S_{0}=0$, $S_{n}=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for $n\in \mathbb{N}$, and let
be the randomly stopped sum of r.v.s $\{\xi _{1},\xi _{2},\dots \}$.
We are interested in conditions under which the d.f. of $S_{\eta }$,
belongs to the class of consistently varying distributions.
Throughout this paper, $f(x)=o(g(x))$ means that $\lim _{x\to \infty }f(x)/g(x)=0$, and $f(x)\sim g(x)$ means that $\lim _{x\to \infty }f(x)/g(x)=1$ for two vanishing (at infinity) functions f and g. Also, we denote the support of a counting r.v. η by
Before discussing the properties of $F_{S_{\eta }}$, we recall the definitions of some classes of heavy-tailed d.f.s, where $\overline{F}(x)=1-F(x)$ for all real x and a d.f. F.
-
• A d.f. F supported on the interval $[0,\infty )$ is subexponential $(F\in \mathcal{S})$ if If a d.f. G is supported on $\mathbb{R}$, then we suppose that G is subexponential $(G\in \mathcal{S})$ if the d.f. $F(x)=G(x)\mathbb{1}_{[0,\infty )}(x)$ satisfies relation (2).
It is known (see, e.g., [4, 11, 13], and Chapters 1.4 and A3 in [8]) that these classes satisfy the following inclusions:
\[ \mathcal{R}\subset \mathcal{C}\subset \mathcal{L}\cap \mathcal{D}\subset \mathcal{S}\subset \mathcal{L}\subset \mathcal{H},\hspace{2em}\mathcal{D}\subset \mathcal{H}.\]
These inclusions are depicted in Fig. 1 with the class $\mathcal{C}$ 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. $F_{S_{\eta }}$ 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]).
Theorem 1.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent copies of a nonnegative r.v. ξ with subexponential d.f. $F_{\xi }$. Let η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. If $\mathbb{E}{(1+\delta )}^{\eta }<\infty $ for some $\delta >0$, then the d.f. $F_{S_{\eta }}\in \mathcal{S}$.
Similar results for the class $\mathcal{D}$ can be found in Leipus and Šiaulys [14]. We present the statement of Theorem 5 from this work.
Theorem 2.
Let $\{\xi _{1},\xi _{2},\dots \}$ be i.i.d. nonnegative r.v.s with common d.f. $F_{\xi }\in \mathcal{D}$ and finite mean $\mathbb{E}\xi $. Let η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$ with d.f. $F_{\eta }$ and finite mean $\mathbb{E}\eta $. Then d.f. $F_{S_{\eta }}\in \mathcal{D}$ iff $\min \{F_{\xi },F_{\eta }\}\in \mathcal{D}$.
We recall only that the d.f. F belongs to the class $\mathcal{D}$ if and only if the upper Matuszewska index ${J_{F}^{+}}<\infty $, where, by definition,
The random convolution closure for the class $\mathcal{L}$ was considered, for instance, in [1, 14, 16, 17]. We now present a particular statement of Theorem 1.1 from [17].
Theorem 3.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent r.v.s, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$ with d.f. $F_{\eta }$. Then the d.f. $F_{S_{\eta }}\in \mathcal{L}$ if the following five conditions are satisfied:
-
(i) $\mathbb{P}(\eta \geqslant \kappa )>0$ for some $\kappa \in \mathbb{N}$;
-
(ii) for all $k\geqslant \kappa $, the d.f. $F_{S_{k}}$ of the sum $S_{k}$ is long tailed;
-
(iii) $\underset{k\geqslant 1}{\sup }\underset{x\in \mathbb{R}}{\sup }(F_{S_{k}}(x)-F_{S_{k}}(x-1))\sqrt{k}<\infty $;
-
(iv) $\underset{z\to \infty }{\limsup }\hspace{0.1667em}\underset{k\geqslant \kappa }{\sup }\hspace{0.1667em}\underset{x\geqslant k(z-1)+z}{\sup }\displaystyle\frac{\overline{F}_{S_{k}}(x-1)}{\overline{F}_{S_{k}}(x)}=1$;
-
(v) $\overline{F}_{\eta }(ax)=o(\sqrt{x}\hspace{0.1667em}\overline{F}_{S_{\kappa }}(x))$ for each $a>0$.
We observe that the case of identically distributed r.v.s is considered in Theorems 1 and 2. In Theorem 3, r.v.s $\{\xi _{1},\xi _{2},\dots \}$ 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].
Theorem 4 ([6], Theorem 2.1).
Let r.v.s $\{\xi _{1},\xi _{2},\dots \}$ be nonnegative independent, not necessarily identically distributed, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. Then the d.f $F_{S_{\eta }}$ belongs to the class $\mathcal{D}$ if the following three conditions are satisfied:
-
(i) $F_{\xi _{\kappa }}\in \mathcal{D}$ for some $\kappa \in \mathrm{supp}(\eta )$,
-
(ii) $\underset{x\to \infty }{\limsup }\underset{n\geqslant \kappa }{\sup }\displaystyle\frac{1}{n\overline{F}_{\xi _{\kappa }}(x)}{\displaystyle\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)<\infty $,
-
(iii) $\mathbb{E}{\eta }^{p+1}<\infty $ for some $p>{J_{F_{\xi _{\kappa }}}^{+}}$.
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 $\mathcal{C}$. If r.v.s $\{\xi _{1},\xi _{2},\dots \}$ are not identically distributed, then different collections of conditions on $\{\xi _{1},\xi _{2},\dots \}$ and η imply that $F_{S_{\eta }}\in \mathcal{C}$. We suppose that some r.v.s from $\{\xi _{1},\xi _{2},\dots \}$ have distributions belonging to the class $\mathcal{C}$, and we find minimal conditions on $\{\xi _{1},\xi _{2},\dots \}$ and η for the distribution of the randomly stopped sum $S_{\eta }$ 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 $S_{\eta }$ 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.
2 Main results
In this section, we present three statements in which we describe the belonging of a randomly stopped sum to the class $\mathcal{C}$. 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 $\{\xi _{1},\xi _{2},\dots \}$ are required, but there is strict requirement for η. Theorem 7 deals with the opposite case: the r.v.s $\{\xi _{1},\xi _{2},\dots \}$ should have finite means, whereas the requirement for η is weaker. It should be noted that the case of real-valued r.v.s $\{\xi _{1},\xi _{2},\dots \}$ is considered in Theorems 5 and 6, whereas Theorem 7 deals with nonnegative r.v.s.
Theorem 5.
Let $\{\xi _{1},\xi _{2},\dots ,\xi _{D}\}$, $D\in \mathbb{N}$, be independent real-valued r.v.s, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots ,\xi _{D}\}$. Then the d.f. $F_{S_{\eta }}$ belongs to the class $\mathcal{C}$ if the following conditions are satisfied:
Theorem 6.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent real-valued r.v.s, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. Then the d.f. $F_{S_{\eta }}$ belongs to the class $\mathcal{C}$ if the following conditions are satisfied:
-
(a) $F_{\xi _{1}}\in \mathcal{C}$,
-
(b) for each $k\geqslant 2$, either $F_{\xi _{k}}\in \mathcal{C}$ or $\overline{F}_{\xi _{k}}(x)=o(\overline{F}_{\xi _{1}}(x))$,
-
(c) $\underset{x\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\displaystyle\frac{1}{n\overline{F}_{\xi _{1}}(x)}{\displaystyle\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)<\infty $,
-
(d) $\mathbb{E}{\eta }^{p+1}<\infty $ for some $p>{J_{F_{\xi _{1}}}^{+}}$.
When $\{\xi _{1},\xi _{2},\dots \}$ are identically distributed with common d.f. $F_{\xi }\in \mathcal{C}$, conditions (a), (b), and (c) of Theorem 6 are satisfied obviously. Hence, we have the following corollary.
Corollary 1 (See also Theorem 3.4 in [3]).
Let $\{\xi _{1},\xi _{2},\dots \}$ be i.i.d. real-valued r.v.s with d.f. $F_{\xi }\in \mathcal{C}$, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. Then the d.f. $F_{S_{\eta }}$ belongs to the class $\mathcal{C}$ if $\mathbb{E}{\eta }^{p+1}<\infty $ for some $p>{J_{F_{\xi }}^{+}}$.
Theorem 7.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent nonnegative r.v.s, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. Then the d.f. $F_{S_{\eta }}$ belongs to the class $\mathcal{C}$ if the following conditions are satisfied:
-
(a) $F_{\xi _{1}}\in \mathcal{C}$,
-
(b) for each $k\geqslant 2$, either $F_{\xi _{k}}\in \mathcal{C}$ or $\overline{F}_{\xi _{k}}(x)=o(\overline{F}_{\xi _{1}}(x))$,
-
(c) $\mathbb{E}\xi _{1}<\infty $,
-
(d) $\overline{F}_{\eta }(x)=o(\overline{F}_{\xi _{1}}(x))$,
-
(e) $\underset{x\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\displaystyle\frac{1}{n\overline{F}_{\xi _{1}}(x)}{\displaystyle\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)<\infty $,
-
(f) $\underset{u\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\displaystyle\frac{1}{n}{\displaystyle\sum \limits_{\begin{array}{c} k=1\\{} \mathbb{E}\xi _{k}\geqslant u\end{array}}^{n}}\mathbb{E}\xi _{k}=0$.
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.
Corollary 2.
Let $\{\xi _{1},\xi _{2},\dots \}$ be i.i.d. nonnegative r.v.s with common d.f. $F_{\xi }\in \mathcal{C}$, and η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$. Then the d.f. $F_{S_{\eta }}$ belongs to the class $\mathcal{C}$ under the following two conditions: $\mathbb{E}\xi <\infty $ and $\overline{F}_{\eta }(x)=o(\overline{F}_{\xi }(x))$.
Further in this section, we present two examples of r.v.s $\{\xi _{1},\xi _{2},\dots \}$ and η for which the random sum $F_{S_{\eta }}$ has a consistently varying tail.
Example 1.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent r.v.s such that $\xi _{k}$ are exponentially distributed for all even k, that is,
\[ \overline{F}_{\xi _{k}}(x)={\mathrm{e}}^{-x},\hspace{1em}x\geqslant 0,\hspace{0.1667em}k\in \{2,4,6,\dots \},\]
whereas, for each odd k, $\xi _{k}$ is a copy of the r.v.
where $\mathcal{U}$ and $\mathcal{G}$ are independent r.v.s, $\mathcal{U}$ is uniformly distributed on the interval $[0,1]$, and $\mathcal{G}$ is geometrically distributed with parameter $q\in (0,1)$, that is,
In addition, let η be a counting r.v. independent of $\{\xi _{1},\xi _{2},\dots \}$ and distributed according to the Poisson law.Theorem 6 implies that the d.f. of the randomly stopped sum $S_{\eta }$ belongs to the class $\mathcal{C}$ because:
-
(b) $F_{\xi _{k}}\in \mathcal{C}$ for $k\in \{3,5,\dots \}$, and $\overline{F}_{\xi _{k}}(x)=o(\overline{F}_{\xi _{1}}(x))$ for $k\in \{2,4,6,\dots \}$,
-
(c) $\underset{x\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\displaystyle\frac{1}{n\overline{F}_{\xi _{1}}(x)}{\displaystyle\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)\leqslant 1$,
-
(d) all moments of the r.v. η are finite.
Note that $\xi _{1}$ does not satisfy condition (c) of Theorem 7 in the case $q\geqslant 1/2$. Hence, Example 1 describes the situation where Theorem 6 should be used instead of Theorem 7.
Example 2.
Let $\{\xi _{1},\xi _{2},\dots \}$ be independent r.v.s such that $\xi _{k}$ are distributed according to the Pareto law (with tail index $\alpha =2$) for all odd k, and $\xi _{k}$ are exponentially distributed (with parameter equal to 1) for all even k, that is,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \overline{F}_{\xi _{k}}(x)=\frac{1}{{x}^{2}},\hspace{1em}x\geqslant 1,\hspace{2.5pt}k\in \{1,3,5,\dots \},\\{} & & \displaystyle \overline{F}_{\xi _{k}}(x)={\mathrm{e}}^{-x},\hspace{1em}x\geqslant 0,\hspace{2.5pt}k\in \{2,4,6,\dots \}.\end{array}\]
In addition, let η be a counting r.v independent of $\{\xi _{1},\xi _{2},\dots \}$ 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 $S_{\eta }$ belongs to the class $\mathcal{C}$ because:
-
(a) $F_{\xi _{1}}\in \mathcal{C}$,
-
(b) $F_{\xi _{k}}\in \mathcal{C}$ for $k\in \{3,5,\dots \}$, and $\overline{F}_{\xi _{k}}(x)=o(\overline{F}_{\xi _{1}}(x))$ for $k\in \{2,4,6,\dots \}$,
-
(c) $\mathbb{E}\xi _{1}=2$,
-
(d) $\overline{F}_{\eta }(x)=o(\overline{F}_{\xi _{1}}(x))$,
-
(e) $\underset{x\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\displaystyle\frac{1}{n\overline{F}_{\xi _{1}}(x)}{\displaystyle\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)\leqslant 1$,
-
(f) $\underset{k\in \mathbb{N}}{\max }\mathbb{E}\xi _{k}=2$.
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.
3 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].
Lemma 2.
Let $\{X_{1},X_{2},\dots X_{n}\}$ be independent real-valued r.v.s. Assume that $\overline{F}_{X_{i}}/\overline{F}(x)\underset{x\to \infty }{\to }b_{i}$ for some subexponential d.f. F and some constants $b_{i}\geqslant 0$, $i\in \{1,2,\dots n\}$. Then
In the next lemma, we show in which cases the convolution $F_{X_{1}}\ast F_{X_{2}}\ast \cdots \ast F_{X_{n}}$ belongs to the class $\mathcal{C}$.
Lemma 3.
Let $\{X_{1},X_{2},\dots ,X_{n}\},n\in \mathbb{N}$, be independent real-valued r.v.s. Then the d.f. $F_{\varSigma _{n}}$ of the sum $\varSigma _{n}=X_{1}+X_{2}+\cdots +X_{n}$ belongs to the class $\mathcal{C}$ if the following conditions are satisfied:
Proof.
Evidently, we can suppose that $n\geqslant 2$. We split our proof into two parts.
First part. Suppose that $F_{X_{k}}\in \mathcal{C}$ for all $k\in \{1,2,\dots ,n\}$. In such a case, the lemma follows from Lemma 1 and the inequality
for $a_{i}\geqslant 0$ and $b_{i}>0$, $i=1,2,\dots ,m$.
(3)
\[ \frac{a_{1}+a_{2}+\cdots +a_{m}}{b_{1}+b_{2}+\cdots +b_{m}}\leqslant \max \bigg\{\frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\dots ,\frac{a_{m}}{b_{m}}\bigg\}\]Namely, using the relation of Lemma 1 and estimate (3), we get that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \underset{x\to \infty }{\limsup }\frac{\overline{F}_{\varSigma _{n}}(xy)}{\overline{F}_{\varSigma _{n}}(x)}& \displaystyle =\underset{x\to \infty }{\limsup }\frac{{\textstyle\sum _{k=1}^{n}}\overline{F}_{X_{k}}(xy)}{{\textstyle\sum _{k=1}^{n}}\overline{F}_{X_{k}}(x)}\\{} & \displaystyle \leqslant \underset{1\leqslant k\leqslant n}{\max }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{X_{k}}(xy)}{\overline{F}_{X_{k}}(x)}\end{array}\]
for arbitrary $y\in (0,1)$.Since $F_{X_{k}}\in \mathcal{C}$ for each k, the last estimate implies that the d.f. $F_{\varSigma _{n}}$ has a consistently varying tail, as desired.
Second part. Now suppose that $F_{X_{k}}\notin \mathcal{C}$ for some of indexes $k\in \{2,3,\dots ,n\}$. By the conditions of the lemma we have that $\overline{F}_{X_{k}}(x)=o(\overline{F}_{X_{1}}(x))$ for such k. Let $\mathcal{K}\subset \{2,3,\dots ,n\}$ be the subset of indexes k such that
for every $y\in (0,1)$.
\[ F_{X_{k}}\notin \mathcal{C}\hspace{1em}\text{and}\hspace{1em}\overline{F}_{X_{k}}(x)=o\big(\overline{F}_{X_{1}}(x)\big).\]
By Lemma 2,
where
Hence,
(4)
\[ \underset{x\to \infty }{\limsup }\frac{\overline{F}_{\widehat{\varSigma }_{n}}(xy)}{\overline{F}_{\widehat{\varSigma }_{n}}(x)}=\underset{x\to \infty }{\limsup }\frac{\overline{F}_{X_{1}}(xy)}{\overline{F}_{X_{1}}(x)}\]Equality (4) implies immediately that the d.f. $F_{\widehat{\varSigma }_{n}}$ belongs to the class $\mathcal{C}$. Therefore, the d.f. $F_{\varSigma _{n}}$ also belongs to the class $\mathcal{C}$ according to the first part of the proof because
and $F_{X_{k}}\in \mathcal{C}$ for each $k\notin \mathcal{K}$. 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 $\mathcal{C}$.
Lemma 4.
Let $\{X_{1},X_{2},\dots \}$ be independent real-valued r.v.s, and $F_{X_{\nu }}\in \mathcal{D}$ for some $\nu \geqslant 1$. Suppose, in addition, that
\[ \underset{x\to \infty }{\limsup }\underset{n\geqslant \nu }{\sup }\frac{1}{n\overline{F}_{X_{\nu }}(x)}{\sum \limits_{i=1}^{n}}\overline{F}_{X_{i}}(x)<\infty .\]
Then, for each $p>{J_{F_{X_{\nu }}}^{+}}$, there exists a positive constant $c_{1}$ such that
for all $n\geqslant \nu $ and $x\geqslant 0$.
In fact, Lemma 4 is proved in [6] for nonnegative r.v.s. However, the lemma remains valid for real-valued r.v.s. To see this, it suffices to observe that $\mathbb{P}(X_{1}+X_{2}+\cdots +X_{n}>x)\leqslant \mathbb{P}({X_{1}^{+}}+{X_{2}^{+}}\cdots +{X_{n}^{+}}>x)$ and $\mathbb{P}(X_{k}>x)=\mathbb{P}({X_{k}^{+}}>x)$, where $n\in \mathbb{N}$, $k\in \{1,2,\dots ,n\}$, $x\geqslant 0$, and ${a}^{+}$ denotes the positive part of a.
Lemma 5.
Let $\{X_{1},X_{2},\dots \}$ be independent real-valued r.v.s, and $F_{X_{\nu }}\in \mathcal{D}$ for some $\nu \geqslant 1$. Let, in addition,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \underset{u\to \infty }{\lim }\underset{n\geqslant \nu }{\sup }\frac{1}{n}{\sum \limits_{k=1}^{n}}\mathbb{E}\big(|X_{k}|\mathbb{1}_{\{X_{k}\leqslant -u\}}\big)=0,\\{} & & \displaystyle \underset{x\to \infty }{\limsup }\underset{n\geqslant \nu }{\sup }\frac{1}{n\overline{F}_{X_{\nu }}(x)}{\sum \limits_{i=1}^{n}}\overline{F}_{X_{i}}(x)<\infty ,\end{array}\]
and $\mathbb{E}X_{k}=\mathbb{E}{X_{k}^{+}}-\mathbb{E}{X_{k}^{-}}=0$ for $k\in \mathbb{N}$. Then, for each $\gamma >0$, there exists a positive constant $c_{2}=c_{2}(\gamma )$ such that
for all $x\geqslant \gamma n$ and all $n\geqslant \nu $.
4 Proofs of the main results
Proof of Theorem 5.
It suffices to prove that
According to estimate (3), for $x>0$ and $y\in (0,1)$, we have
(6)
\[ \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{S_{\eta }}(xy)}{\overline{F}_{S_{\eta }}(x)}\leqslant 1.\]
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \frac{\overline{F}_{S_{\eta }}(xy)}{\overline{F}_{S_{\eta }}(x)}& \displaystyle =\frac{{\textstyle\sum _{\begin{array}{c} n=1\\{} n\in \mathrm{supp}(\eta )\end{array}}^{D}}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{{\textstyle\sum _{\begin{array}{c} n=1\\{} n\in \mathrm{supp}(\eta )\end{array}}^{D}}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\leqslant \underset{\begin{array}{c} 1\leqslant n\leqslant D\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\frac{\mathbb{P}(S_{n}>xy)}{\mathbb{P}(S_{n}>x)}.\end{array}\]
Hence, by Lemma 3,
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{S_{\eta }}(xy)}{\overline{F}_{S_{\eta }}(x)}& \displaystyle \leqslant \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\underset{\begin{array}{c} 1\leqslant n\leqslant D\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\frac{\overline{F}_{S_{n}}(xy)}{\overline{F}_{S_{n}}(x)}\\{} & \displaystyle \leqslant \underset{\begin{array}{c} 1\leqslant n\leqslant D\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{S_{n}}(xy)}{\overline{F}_{S_{n}}(x)}=1,\end{array}\]
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 $K\in \mathbb{N}$ and $x>0$, we have
\[ \mathbb{P}(S_{\eta }>x)=\Bigg({\sum \limits_{n=1}^{K}}+{\sum \limits_{n=K+1}^{\infty }}\Bigg)\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n).\]
Therefore, for $x>0$ and $y\in (0,1)$, we have
(7)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \frac{\mathbb{P}(S_{\eta }>xy)}{\mathbb{P}(S_{\eta }>x)}& \displaystyle =\frac{{\textstyle\sum _{n=1}^{K}}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle \hspace{1em}+\hspace{2.5pt}\frac{{\textstyle\sum _{n=K+1}^{\infty }}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle =:\mathcal{J}_{1}+\mathcal{J}_{2}.\end{array}\]The random variable η is not degenerate at zero, so there exists $a\in \mathbb{N}$ such that $\mathbb{P}(\eta =a)>0$. If $K\geqslant a$, then using inequality (3), we get
Since $\mathcal{C}\subset \mathcal{D}$, we can use Lemma 4 for the numerator of $\mathcal{J}_{2}$ to obtain
\[ \mathcal{J}_{1}\leqslant \frac{{\textstyle\sum _{\begin{array}{c} n=1\\{} n\in \mathrm{supp}(\eta )\end{array}}^{K}}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{{\textstyle\sum _{\begin{array}{c}n=1\\{}n\in \mathrm{supp}(\eta )\end{array}}^{K}}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\leqslant \underset{\begin{array}{c} 1\leqslant n\leqslant K\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\frac{\mathbb{P}(S_{n}>xy)}{\mathbb{P}(S_{n}>x)}.\]
Similarly as in the proof of Theorem 5, it follows that
(8)
\[ \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\mathcal{J}_{1}\leqslant \underset{\begin{array}{c}1\leqslant n\leqslant K\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{S_{n}}(xy)}{\overline{F}_{S_{n}}(x)}=1.\]
\[ {\sum \limits_{n=K+1}^{\infty }}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)\leqslant c_{3}\overline{F}_{\xi _{1}}(xy){\sum \limits_{n=K+1}^{\infty }}{n}^{p+1}\mathbb{P}(\eta =n)\]
with some positive constant $c_{3}$. For the denominator of $\mathcal{J}_{2}$, we have that
The conditions of the theorem imply that
\[ S_{a}=\xi _{1}+\sum \limits_{k\in \mathcal{K}_{a}}\xi _{k}+\sum \limits_{k\notin \mathcal{K}_{a}}\xi _{k},\]
where $\mathcal{K}_{a}=\{k\in \{2,\dots ,a\}:F_{\xi _{k}}\notin \mathcal{C},\overline{F}_{\xi _{k}}(x)=o(\hspace{0.1667em}\overline{F}_{\xi _{1}}(x))\}$.By Lemma 2
where $F_{\widehat{S}_{a}}$ is the d.f. of the sum
In addition, by Lemma 3 we have that the d.f. $F_{\widehat{S}_{a}}$ belongs to the class $\mathcal{C}$.
If $k\notin \mathcal{K}_{a}$, then $F_{\xi _{k}}\in \mathcal{C}$ by the conditions of the theorem. This fact and Lemma 1 imply that
Estimates (7), (8), and (10) imply that
\[ \underset{x\to \infty }{\liminf }\frac{\mathbb{P}(S_{a}>x)}{\overline{F}_{\xi _{1}}(x)}\geqslant 1+\sum \limits_{k\notin \mathcal{K}_{a}}\underset{x\to \infty }{\liminf }\frac{\overline{F}_{\xi _{k}}(x)}{\overline{F}_{\xi _{1}}(x)}.\]
Hence,
for x sufficiently large. Therefore,
(10)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\mathcal{J}_{2}\\{} & \displaystyle \hspace{1em}\leqslant \frac{2\hspace{0.1667em}c_{3}}{\mathbb{P}(\eta =a)}\bigg(\underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\overline{F}_{\xi _{1}}(xy)}{\overline{F}_{\xi _{1}}(x)}\bigg){\sum \limits_{n=K+1}^{\infty }}{n}^{p+1}\mathbb{P}(\eta =n).\end{array}\]
\[ \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\mathbb{P}(S_{\eta }>xy)}{\mathbb{P}(S_{\eta }>x)}\leqslant 1+\frac{2\hspace{0.1667em}c_{3}}{\mathbb{P}(\eta =a)}\mathbb{E}{\eta }^{p+1}\mathbb{1}_{\{\eta >K\}}\]
for arbitrary $K\geqslant a$.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
for a positive constant $c_{6}$ and all $n\in \mathbb{N}$.
\[ {\sum \limits_{i=1}^{n}}\overline{F}_{\xi _{i}}(x)\leqslant c_{5}n\overline{F}_{\xi _{1}}(x),\hspace{1em}x\geqslant c_{4},\hspace{2.5pt}n\in \mathbb{N}.\]
Therefore,
(11)
\[ \mathbb{E}S_{n}={\sum \limits_{j=1}^{n}}\mathbb{E}\xi _{j}={\sum \limits_{j=1}^{n}}\Bigg(\hspace{0.1667em}{\int _{0}^{c_{4}}}+{\int _{c_{4}}^{\infty }}\hspace{0.1667em}\Bigg)\overline{F}_{\xi _{j}}(u)du\leqslant c_{4}n+c_{5}n\mathbb{E}\xi _{1}=:c_{6}n\]If $K\in \mathbb{N}$ and $x>4Kc_{6}$, then we have
if $xy>4Kc_{6}$, $x>0$, and $y\in (0,1)$.
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{P}(S_{\eta }>x)& \displaystyle =\mathbb{P}(S_{\eta }>x,\eta \leqslant K)\\{} & \displaystyle \hspace{1em}+\mathbb{P}\bigg(S_{\eta }>x,K<\eta \leqslant \frac{x}{4c_{6}}\bigg)\\{} & \displaystyle \hspace{1em}+\mathbb{P}\bigg(S_{\eta }>x,\eta >\frac{x}{4c_{6}}\bigg).\end{array}\]
Therefore,
(12)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \frac{\mathbb{P}(S_{\eta }>xy)}{\mathbb{P}(S_{\eta }>x)}& \displaystyle =\frac{\mathbb{P}(S_{\eta }>xy,\eta \leqslant K)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle \hspace{1em}+\frac{\mathbb{P}\big(S_{\eta }>xy,K<\eta \leqslant \frac{xy}{4c_{6}}\big)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle \hspace{1em}+\frac{\mathbb{P}\big(S_{\eta }>xy,\eta >\frac{xy}{4c_{6}}\big)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle =:\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}\end{array}\]The random variable η is not degenerate at zero, so $\mathbb{P}(\eta =a)>0$ for some $a\in \mathbb{N}$. If $K\geqslant a$, then
similarly to estimate (8) in Theorem 6.
(13)
\[ \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\mathcal{I}_{1}\leqslant 1\]For the numerator of $\mathcal{I}_{2}$, we have
by inequality (11).
(14)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathcal{I}_{2,1}:=& \displaystyle \mathbb{P}\bigg(S_{\eta }>xy,K<\eta \leqslant \frac{xy}{4c_{6}}\bigg)\\{} \displaystyle =& \displaystyle \sum \limits_{K<n\leqslant \frac{xy}{4c_{6}}}\mathbb{P}\Bigg({\sum \limits_{i=1}^{n}}(\xi _{i}-\mathbb{E}\xi _{i})>xy-{\sum \limits_{j=1}^{n}}\mathbb{E}\xi _{j}\Bigg)\mathbb{P}(\eta =n)\\{} \displaystyle \leqslant & \displaystyle \sum \limits_{K<n\leqslant \frac{xy}{4c_{6}}}\mathbb{P}\Bigg({\sum \limits_{i=1}^{n}}(\xi _{i}-\mathbb{E}\xi _{i})>\frac{3}{4}xy\Bigg)\mathbb{P}(\eta =n)\end{array}\]The random variables $\xi _{1}-\mathbb{E}\xi _{1}$, $\xi _{2}-\mathbb{E}\xi _{2},\dots \hspace{0.1667em}$ satisfy the conditions of Lemma 5. Namely, $\mathbb{E}(\xi _{k}-\mathbb{E}\xi _{k})=0$ for $k\in \mathbb{N}$ and $F_{\xi _{1}-\mathbb{E}\xi _{1}}\in \mathcal{C}\subset \mathcal{D}$ obviously. In addition,
since the r.v.s $\{\xi _{1},\xi _{2},\dots \}$ are nonnegative by assumption. Hence,
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{x\to \infty }{\limsup }\hspace{0.1667em}\underset{n\geqslant 1}{\sup }\frac{1}{n\hspace{0.1667em}\mathbb{P}(\xi _{1}-\mathbb{E}\xi _{1}>x)}{\sum \limits_{k=1}^{n}}\mathbb{P}(\xi _{i}-\mathbb{E}\xi _{i}>x)<\infty \end{array}\]
by conditions (a), (c) and (e). Finally,
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{u\to \infty }{\limsup }\hspace{0.1667em}\underset{n\geqslant 1}{\sup }\frac{1}{n}{\sum \limits_{k=1}^{n}}\mathbb{E}(|\xi _{k}-\mathbb{E}\xi _{k}|\mathbb{1}_{\{\xi _{k}-\mathbb{E}\xi _{k}\leqslant -u\}})\\{} & \displaystyle \hspace{1em}=\underset{u\to \infty }{\limsup }\hspace{0.1667em}\underset{n\geqslant 1}{\sup }\frac{1}{n}{\sum \limits_{k=1}^{n}}\mathbb{E}\big((\mathbb{E}\xi _{k}-\xi _{k})\mathbb{1}_{\{\xi _{k}-\mathbb{E}\xi _{k}\leqslant -u\}}\big)\\{} & \displaystyle \leqslant \underset{u\to \infty }{\limsup }\underset{n\geqslant 1}{\sup }\frac{1}{n}\sum \limits_{\begin{array}{c}1\leqslant k\leqslant n\\{}\mathbb{E}\xi _{k}\geqslant u\end{array}}\mathbb{E}\xi _{k}=0\end{array}\]
because of condition (f). So, applying the estimate of Lemma 5 to (14), we get
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathcal{I}_{2,1}& \displaystyle \leqslant c_{7}\sum \limits_{K<n\leqslant \frac{xy}{4c_{6}}}n\overline{F}_{\xi _{1}}\bigg(\frac{3}{4}xy+\mathbb{E}\xi _{1}\bigg)\mathbb{P}(\eta =n)\\{} & \displaystyle \leqslant \hspace{2.5pt}c_{7}\overline{F}_{\xi _{1}}\Big(\hspace{0.1667em}\frac{3}{4}xy\Big)\mathbb{E}\eta \mathbb{1}_{\{\eta >K\}}\end{array}\]
with a positive constant $c_{7}$. For the denominator of $\mathcal{I}_{2}$, we can use the inequality
(15)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{P}(S_{\eta }>x)& \displaystyle ={\sum \limits_{n=1}^{\infty }}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)\\{} & \displaystyle \geqslant {\sum \limits_{n=1}^{\infty }}\mathbb{P}(\xi _{1}>x)\mathbb{P}(\eta =n)\\{} & \displaystyle \geqslant \overline{F}_{\xi _{1}}(x)\mathbb{P}(\eta =a)\end{array}\]If $y\in (1/2,1)$, then the last estimate implies that
with some positive constant $c_{8}$ because $F_{\xi _{1}}\in \mathcal{C}\subset \mathcal{D}$.
(16)
\[ \underset{x\to \infty }{\limsup }\mathcal{I}_{2}\leqslant \frac{c_{7}}{\mathbb{P}(\eta =a)}\mathbb{E}\eta \mathbb{1}_{\{\eta >K\}}\underset{x\to \infty }{\limsup }\hspace{2.5pt}\frac{\overline{F}_{\xi _{1}}\big(\frac{3}{8}x\big)}{\overline{F}_{\xi _{1}}(x)}\leqslant c_{8}\mathbb{E}\eta \mathbb{1}_{\{\eta >K\}}\]Using inequality (15) again, we obtain
by condition (d).
\[ \mathcal{I}_{3}\leqslant \frac{\mathbb{P}\big(\eta >\frac{xy}{4c_{6}}\big)}{\mathbb{P}(S_{\eta }>x)}\leqslant \frac{1}{\mathbb{P}(\eta =a)}\frac{\overline{F}_{\eta }\big(\frac{xy}{4c_{6}}\big)}{\overline{F}_{\xi _{1}}\big(\frac{xy}{4c_{6}}\big)}\frac{\overline{F}_{\xi _{1}}\big(\frac{xy}{4c_{6}}\big)}{\overline{F}_{\xi _{1}}(x)}.\]
Therefore, for $y\in (1/2,1)$, we get
(17)
\[ \underset{x\to \infty }{\limsup }\mathcal{I}_{3}\leqslant \frac{1}{\mathbb{P}(\eta =a)}\underset{x\to \infty }{\limsup }\frac{\overline{F}_{\eta }\big(\frac{xy}{4c_{6}}\big)}{\overline{F}_{\xi _{1}}\big(\frac{xy}{4c_{6}}\big)}\underset{x\to \infty }{\limsup }\frac{\overline{F}_{\xi _{1}}\big(\frac{xy}{4c_{6}}\big)}{\overline{F}_{\xi _{1}}(x)}=0\]Estimates (12), (13), (16), and (17) imply that
\[ \underset{y\uparrow 1}{\limsup }\underset{x\to \infty }{\limsup }\frac{\mathbb{P}(S_{\eta }>xy)}{\mathbb{P}(S_{\eta }>x)}\leqslant 1+c_{8}\mathbb{E}\eta \mathbb{1}_{\{\eta >K\}}\]
for $K\geqslant a$.Letting K tend to infinity, we get the desired estimate (6) because $\mathbb{E}\eta <\infty $ by conditions (c) and (d). The theorem is proved. □
