VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA52 10.15559/16-VMSTA52 Research Article Random convolution of inhomogeneous distributions with O -exponential tail DanilenkoSvetlanasvetlana.danilenko@vgtu.lta PaškauskaitėSimonasimona.paskauskaite@mif.vu.stud.ltb ŠiaulysJonasjonas.siaulys@mif.vu.ltb∗ Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius LT-10223, Lithuania Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania Corresponding author. 2016 442016317994 2812016 2132016 2132016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

Let {ξ1,ξ2,…} be a sequence of independent random variables (not necessarily identically distributed), and η be a counting random variable independent of this sequence. We obtain sufficient conditions on {ξ1,ξ2,…} and η under which the distribution function of the random sum Sη=ξ1+ξ2+⋯+ξη belongs to the class of O -exponential distributions.

 Heavy tail exponential tail O-exponential tail random sum random convolution inhomogeneous distributions closure property 62E20 60E05 60F10 44A35 Introduction

Let {ξ1,ξ2,…} be a sequence of independent random variables (r.v.s) with distribution functions (d.f.s) {Fξ1,Fξ2,…} , 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 {ξ1,ξ2,…} are independent. Let S0=0 and Sn=ξ1+ξ2+⋯+ξn , n∈N , be the partial sums, and let Sη= ∑k=1ηξk be the random sum of {ξ1,ξ2,…} .

We are interested in conditions under which the d.f. of Sη FSη(x)=P(Sη⩽x)= ∑n=0∞P(η=n)P(Sn⩽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 0<lim infx→∞F‾(x+a)F‾(x)⩽lim supx→∞F‾(x+a)F‾(x)<∞ for all a∈R , where F‾(x)=1−F(x) , x∈R , is the tail of a d.f. F.

Note that if F∈OL , then F‾(x)>0 0$]]> for all x∈R .

It is obvious that a d.f. F belongs to the class OL if and only if lim supx→∞F‾(x−1)F‾(x)<∞ or, equivalently, if and only if supx⩾0F‾(x−1)F‾(x)<∞.

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 0$]]>, we have limx→∞F‾(x+a)F‾(x)=1.

A d.f. F is said to belong to the class L(γ) of exponential distributions with some γ>0 0$]]> if for any fixed a>0 0$]]>, we have limx→∞F‾(x+a)F‾(x)=e−aγ.

A d.f. F belongs to the class D (or has a dominatingly varying tail) if for every fixed a∈(0,1) , we have lim supx→∞F‾(xa)F‾(x)<∞.

A d.f. F supported on the interval [0,∞) belongs to the class S (or is subexponential) if limx→∞F∗F‾(x)F‾(x)=2, 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 μ:=∫[0,∞)xdF(x)<∞and ∫0xF‾(x−y)F‾(y)dy∼x→∞2μF‾(x).

If a d.f. F is supported on R , then F belongs to some of the classes S or S∗ if F+(x)=F(x)1{[0,∞)}(x) belongs to the corresponding class.

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 S∗⊂S⊂L⊂OL,D⊂OL,⋃γ>0L(γ)⊂OL. 0}\mathcal{L}(\gamma )\subset \mathcal{OL}.\]]]>

Now we present a few known results on when the d.f. FSη 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]).

Let {ξ1,ξ2,…} be independent copies of a nonnegative r.v. ξ with subexponential d.f. Fξ . Let η be a counting r.v. independent of {ξ1,ξ2,…} . If E(1+δ)η<∞ for some δ>0 0$]]>, then FSη∈S .

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].

Let {ξ1,ξ2,…} be independent copies of a nonnegative r.v. ξ with strongly subexponential d.f. Fξ and finite mean Eξ . Let η be a counting r.v. independent of {ξ1,ξ2,…} . If P(η>x/c)=x→∞o(F‾ξ(x)) x/c)\underset{x\to \infty }{=}o(\overline{F}_{\xi }(x))$]]> for some c>Eξ \mathbb{E}\xi $]]>, then FSη∈S∗ .

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].

Let {ξ1,ξ2,…} be independent r.v.s with common d.f. Fξ∈L , and let η be a counting r.v. independent of {ξ1,ξ2,…} having d.f.  Fη . If F‾η(δx)=x→∞o(xF‾ξ(x)) for each δ∈(0,1) , then FSη∈L .

In all presented results, r.v.s {ξ1,ξ2,…} 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 {ξ1,ξ2,…} may be not identically distributed, then various collections of conditions on r.v.s {ξ1,ξ2,…} and η imply that FSη∈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 Fξk for large indices k are uniformly comparable with itself at the points x and x−1 for all x∈[0,∞) .

Let {ξ1,ξ2,…} be independent nonnegative random variables with d.f.s {Fξ1,Fξ2,…} , and let η be a counting r.v. independent of {ξ1,ξ2,…} . Then FSη∈OL if the following three conditions are satisfied.

For some κ∈supp(η)∖{0}={n∈N:P(η=n)>0} 0\}$]]>, Fξκ∈OL .

For each k∈supp(η) , k⩽κ , either limx→∞F‾ξk(x)F‾ξκ(x)=0 or Fξk∈OL .

supx⩾0supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)<∞ .

Since each d.f. from the class OL is comparable with itself, the next assertion follows immediately from Theorem 4.

Let {ξ1,ξ2,…} be independent nonnegative random variables with common d.f. Fξ∈OL . Then the d.f. of random sum FSη is O -exponential for an arbitrary counting r.v. η.

Our second main assertion is dealt with counting r.v.s having finite support.

Let {ξ1,ξ2,…,ξD} , D∈N , be independent nonnegative random variables with d.f.s {Fξ1,Fξ2,…FξD} , and let η be a counting r.v. independent of {ξ1,ξ2,…,ξD} . Then FSη∈OL under the following three conditions.

P(η⩽D)=1 .

For some κ∈supp(η)∖{0} , Fξκ∈OL .

For each k∈{1,2,…,D} , either limx→∞F‾ξk(x)F‾ξκ(x)=0 or Fξk∈OL .

Our last main assertion describes the case where the tails of d.f.s Fξk 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.

Let {ξ1,ξ2,…} be independent nonnegative random variables with d.f.s {Fξ1,Fξ2,…} , and let η be a counting r.v. d.f. Fη independent of {ξ1,ξ2,…} . Then FSη∈OL if the following five conditions are satisfied.

For some κ∈supp(η)∖{0} , Fξκ∈OL .

For each k∈supp(η) , k⩽κ , either limx→∞F‾ξk(x)F‾ξκ(x)=0 or Fξk∈OL .

lim supx→∞supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)<∞ .

lim supk→∞1k∑l=1ksupx⩾0(F‾ξκ+l(x−1)−F‾ξκ+l(x))<1 .

For each δ∈(0,1) , F‾η(δx)=O(xF‾ξκ(x)) .

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)).

Let F and G be two d.f.s satisfying F‾(x)>0 0$]]>, G‾(x)>0 0$]]>, x∈R . Then F∗G‾(x−t)F∗G‾(x)⩽max{supy⩾vF‾(y−t)F‾(y),supy⩾x−v+tG‾(y−t)G‾(y)} for all x∈R , v∈R , and t>0 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 QX(λ)=supx∈RP(x⩽X⩽x+λ),λ∈[0,∞).

The proof of the next lemma can be found in [14] (Theorem 2.15).

Let X1,X2,…,Xn be independent r.v.s, and let Zn=∑k=1nXk . Then, for all n∈N , QZn(λ)⩽Aλ{ ∑k=1nλk2(1−QXk(λk))}−1/2, where A is an absolute constant, and 0<λk⩽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 .

Let X1 and X2 be independent r.v.s with d.f.s FX1 and FX2 , respectively. Then the d.f. FX1∗FX2 of the sum X1+X2 is O -exponential if FX1∈OL and one of the following two conditions holds: ∙limx→∞F‾X2(x)F‾X1(x)=0,∙FX2∈OL.

We split the proof into three parts.

I. First, suppose that P(X2⩽D)=1 for some D>0 0$]]>. In this case, condition (2) holds evidently.

For each real x, we have FX1∗FX2‾(x)=P(X1+X2>x)=∫(−∞,D]F‾X1(x−y)dFX2(y). x)=\underset{(-\infty ,D]}{\int }\overline{F}_{X_{1}}(x-y)\text{d}F_{X_{2}}(y).\end{array}\]]]> Hence, for such x, FX1∗FX2‾(x−1)FX1∗FX2‾(x)=∫(−∞,D]F‾X1(x−1−y)F‾X1(x−y)F‾X1(x−y)dFX2(y)∫(−∞,D]F‾X1(x−y)dFX2(y)⩽∫(−∞,D]supy⩽DF‾X1(x−1−y)F‾X1(x−y)F‾X1(x−y)dFX2(y)∫(−∞,D]F‾X1(x−y)dFX2(y)=supz⩾x−DF‾X1(z−1)F‾X1(z). This estimate implies that lim supx→∞FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽lim supx→∞supz⩾x−DF‾X1(z−1)F‾X1(z)=lim supy→∞F‾X1(y−1)F‾X1(y)<∞ because FX1∈OL . So, FX1∗FX2∈OL as well.

II. Now let us consider the case where condition (2) holds but F‾X2(x)>0 0$]]> for all x∈R . For each real x, we have FX1∗FX2‾(x)= ∫−∞∞FX1‾(x−y)dFX2(y). Therefore, FX1∗FX2‾(x−1)=(∫(−∞,x−M]+∫(x−M,∞))F‾X1(x−1−y)dFX2(y)⩽∫(−∞,x−M]F‾X1(x−1−y)F‾X1(x−y)F‾X1(x−y)dFX2(y)+F‾X2(x−M)⩽supz⩾MF‾X1(z−1)F‾X1(z)∫(−∞,x−M]F‾X1(x−y)dFX2(y)+F‾X2(x−M) for all M,x such that 0<M<x−1 . In addition, for such M and x, we obtain FX1∗FX2‾(x)⩾∫(−∞,x−M]F‾X1(x−y)dFX2(y),FX1∗FX2‾(x)⩾∫(M,∞)F‾X1(x−y)dFX2(y)⩾F‾X1(x−M)F‾X2(M). The obtained estimates imply that FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽supz⩾MF‾X1(z−1)F‾X1(z)+F‾X2(x−M)F‾X1(x−M)F‾X2(M) for all x and M such that 0<M<x−1 . Consequently, lim supx→∞FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽supz⩾MF‾X1(z−1)F‾X1(z)+1F‾X2(M)lim supx→∞F‾X2(x−M)F‾X1(x−M)=supz⩾MF‾X1(z−1)F‾X1(z) for all positive M. Therefore, lim supx→∞FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽lim supM→∞F‾X1(M−1)F‾X1(M)<∞ because FX1 is O -exponential. Consequently, FX1∗FX2∈OL by (1).

III. It remains to prove the assertion when both d.f.s FX1 and FX2 are O -exponential. By Lemma 1 we have FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽max{supz⩾MF‾X1(z−1)F‾X1(z),supz⩾x−M+1F‾X2(z−1)F‾X2(z)} for all x and M such that 0<M<x−1 . Therefore, for every positive M, lim supx→∞FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽max{supz⩾MF‾X1(z−1)F‾X1(z),lim supx→∞supz⩾x−M+1F‾X2(z−1)F‾X2(z)}=max{supz⩾MF‾X1(z−1)F‾X1(z),lim supy→∞F‾X2(y−1)F‾X2(y)}. Letting M tend to infinity, we get that lim supx→∞FX1∗FX2‾(x−1)FX1∗FX2‾(x)⩽max{lim supM→∞F‾X1(M−1)F‾X1(M),lim supy→∞F‾X2(y−1)F‾X2(y)}<∞ because FX1 and FX2 belong to class OL . Consequently, FX1∗FX2∈OL due to requirement (1). Lemma 3 is proved.  □

Let {X1,X2,…,Xn} be independent nonnegative r.v.s with d.f.s {FX1,FX2,…,FXn} . Let FX1∈OL and suppose that, for each k∈{2,3,…,n} , either limx→∞F‾Xk(x)F‾X1(x)=0 or FXk∈OL . Then the d.f. FX1∗FX2∗⋯∗FXn belongs to the class OL .

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, FX1∗FX2∗⋯∗FXm∈OL , and we will show that the statement is correct for n=m+1 .

Conditions of the lemma imply that FXm+1∈OL or limx→∞F‾Xm+1(x)FX1∗FX2∗⋯∗FXm‾(x)=limx→∞F‾Xm+1(x)P(X1+⋯+Xm>x)⩽limx→∞F‾Xm+1(x)P(X1>x)=limx→∞F‾Xm+1(x)F‾X1(x)=0. x)}\\{} & \displaystyle \hspace{1em}\leqslant \underset{x\to \infty }{\lim }\frac{\overline{F}_{X_{m+1}}(x)}{\mathbb{P}(X_{1}>x)}=\underset{x\to \infty }{\lim }\frac{\overline{F}_{X_{m+1}}(x)}{\overline{F}_{X_{1}}(x)}=0.\end{array}\]]]> So, using Lemma 3 again, we get FX1∗FX2∗⋯∗FXm+1=(FX1∗FX2∗⋯∗FXm)∗FXm+1∈OL. 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. FSκ(x)=P(Sκ⩽x) belongs to the class OL . So, we have lim supx→∞F‾Sκ(x−1)F‾Sκ(x)<∞ or, equivalently, supx⩾0F‾Sκ(x−1)F‾Sκ(x)⩽c1 for some positive constant c1 .

We observe that, for all x⩾0 , P(Sη>x−1)P(Sη>x)=J1(x)+J2(x), x-1)}{\mathbb{P}(S_{\eta }>x)}=\mathcal{J}_{1}(x)+\mathcal{J}_{2}(x),\]]]> where J1(x)=P(Sη>x−1,η⩽κ)P(Sη>x),J2(x)=P(Sη>x−1,η>κ)P(Sη>x). x-1,\eta \leqslant \kappa )}{\mathbb{P}(S_{\eta }>x)},\\{} \displaystyle \mathcal{J}_{2}(x)=\frac{\mathbb{P}(S_{\eta }>x-1,\eta >\kappa )}{\mathbb{P}(S_{\eta }>x)}.\end{array}\]]]> Since κ∈supp(η) , we obtain J1(x)=∑n=0κP(Sn>x−1)P(η=n)∑n=0∞P(Sn>x)P(η=n)⩽1P(Sκ>x)P(η=κ) ∑n=0κP(Sκ>x−1)P(η=n)=P(Sκ>x−1)P(Sκ>x)P(η⩽κ)P(η=κ). x-1)\mathbb{P}(\eta =n)}{{\textstyle\sum _{n=0}^{\infty }}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\\{} & \displaystyle \leqslant \frac{1}{\mathbb{P}(S_{\kappa }>x)\mathbb{P}(\eta =\kappa )}{\sum \limits_{n=0}^{\kappa }}\mathbb{P}(S_{\kappa }>x-1)\mathbb{P}(\eta =n)\\{} & \displaystyle =\frac{\mathbb{P}(S_{\kappa }>x-1)}{\mathbb{P}(S_{\kappa }>x)}\frac{\mathbb{P}(\eta \leqslant \kappa )}{\mathbb{P}(\eta =\kappa )}.\end{array}\]]]> Hence, it follows from (3) that lim supx→∞J1(x)<∞. By Lemma 1 we have P(Sκ+1>x−1)P(Sκ+1>x)⩽max{supz⩾MP(Sκ>z−1)P(Sκ>z),supz⩾x−M+1F‾ξκ+1(z−1)F‾ξκ+1(z)} x-1)}{\mathbb{P}(S_{\kappa +1}>x)}\leqslant \max \bigg\{\underset{z\geqslant M}{\sup }\frac{\mathbb{P}(S_{\kappa }>z-1)}{\mathbb{P}(S_{\kappa }>z)},\underset{z\geqslant x-M+1}{\sup }\frac{\overline{F}_{\xi _{\kappa +1}}(z-1)}{\overline{F}_{\xi _{\kappa +1}}(z)}\bigg\}\]]]> for all real x and M.

The third condition of the theorem implies that supx⩾0F‾ξκ+k(x−1)F‾ξκ+k(x)⩽c2 for all k∈N and some positive c2 .

If we choose M=x/2 in estimate (7), then, using (4), we get supx⩾0P(Sκ+1>x−1)P(Sκ+1>x)⩽maxc1,c2:=c3. x-1)}{\mathbb{P}(S_{\kappa +1}>x)}\leqslant \max \left\{c_{1},c_{2}\right\}:=c_{3}.\]]]> Applying Lemma 1 again, we obtain P(Sκ+2>x−1)P(Sκ+2>x)⩽max{supz⩾MP(Sκ+1>z−1)P(Sκ+1>z),supz⩾x−M+1F‾ξκ+2(z−1)F‾ξκ+2(z)}. x-1)}{\mathbb{P}(S_{\kappa +2}>x)}\leqslant \max \bigg\{\underset{z\geqslant M}{\sup }\frac{\mathbb{P}(S_{\kappa +1}>z-1)}{\mathbb{P}(S_{\kappa +1}>z)},\underset{z\geqslant x-M+1}{\sup }\frac{\overline{F}_{\xi _{\kappa +2}}(z-1)}{\overline{F}_{\xi _{\kappa +2}}(z)}\bigg\}.\]]]> By choosing M=x/2 we get from inequalities (8) and (9) that supx⩾0P(Sκ+2>x−1)P(Sκ+2>x)⩽c3. x-1)}{\mathbb{P}(S_{\kappa +2}>x)}\leqslant c_{3}.\]]]> Continuing the process, we find supx⩾0P(Sκ+k>x−1)P(Sκ+k>x)⩽c3 x-1)}{\mathbb{P}(S_{\kappa +k}>x)}\leqslant c_{3}\]]]> for all k∈N . Therefore, J2(x)=1P(Sη>x) ∑k=1∞P(Sκ+k>x−1)P(η=κ+k)⩽c3P(Sη>x) ∑k=1∞P(Sκ+k>x)P(η=κ+k)⩽c3P(Sη>x)P(Sη>x)=c3 x)}{\sum \limits_{k=1}^{\infty }}\mathbb{P}(S_{\kappa +k}>x-1)\mathbb{P}(\eta =\kappa +k)\\{} & \displaystyle \leqslant \frac{c_{3}}{\mathbb{P}(S_{\eta }>x)}{\sum \limits_{k=1}^{\infty }}\mathbb{P}(S_{\kappa +k}>x)\mathbb{P}(\eta =\kappa +k)\\{} & \displaystyle \leqslant \frac{c_{3}\mathbb{P}(S_{\eta }>x)}{\mathbb{P}(S_{\eta }>x)}=c_{3}\end{array}\]]]> for all x⩾0 .

The obtained relations (5), (6), and (10) imply that lim supx→∞P(Sη>x−1)P(Sη>x)<∞. x-1)}{\mathbb{P}(S_{\eta }>x)}<\infty .\]]]> Therefore, the d.f. FSη belongs to the class OL due to requirement (1). Theorem 4 is proved. □

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 Sk=ξκ+∑n=1,n≠κkξn for each k⩾κ . Hence, by Lemma 4, FSk∈OL for all κ⩽k⩽D .

If x⩾1 , then we have P(Sη>x−1)P(Sη>x)=∑n=1n∈supp(η)DP(Sn>x−1)P(η=n)∑n=1n∈supp(η)DP(Sn>x)P(η=n)⩽P(Sκ>x−1)P(η⩽κ)+∑n=κ+1n∈supp(η)DP(Sn>x−1)P(η=n)P(Sκ>x)P(η=κ)+∑n=κ+1n∈supp(η)DP(Sn>x)P(η=n)⩽max{P(Sκ>x−1)P(η⩽κ)P(Sκ>x)P(η=κ),maxκ+1⩽n⩽Dn∈supp(η)P(Sn>x−1)P(Sn>x)}, x-1)}{\mathbb{P}(S_{\eta }>x)}& \displaystyle =\frac{{\textstyle\sum _{\genfrac{}{}{0pt}{}{n=1}{n\in \mathrm{supp}(\eta )}}^{D}}\mathbb{P}(S_{n}>x-1)\mathbb{P}(\eta =n)}{{\textstyle\sum _{\genfrac{}{}{0pt}{}{n=1}{n\in \mathrm{supp}(\eta )}}^{D}}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\\{} & \displaystyle \leqslant \frac{\mathbb{P}(S_{\kappa }>x-1)\mathbb{P}(\eta \leqslant \kappa )+{\textstyle\sum _{\genfrac{}{}{0pt}{}{n=\kappa +1}{n\in \mathrm{supp}(\eta )}}^{D}}\mathbb{P}(S_{n}>x-1)\mathbb{P}(\eta =n)}{\mathbb{P}(S_{\kappa }>x)\mathbb{P}(\eta =\kappa )+{\textstyle\sum _{\genfrac{}{}{0pt}{}{n=\kappa +1}{n\in \mathrm{supp}(\eta )}}^{D}}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\\{} & \displaystyle \leqslant \max \bigg\{\frac{\mathbb{P}(S_{\kappa }>x-1)\mathbb{P}(\eta \leqslant \kappa )}{\mathbb{P}(S_{\kappa }>x)\mathbb{P}(\eta =\kappa )},\underset{\genfrac{}{}{0pt}{}{\kappa +1\leqslant n\leqslant D}{n\in \mathrm{supp}(\eta )}}{\max }\frac{\mathbb{P}(S_{n}>x-1)}{\mathbb{P}(S_{n}>x)}\bigg\},\end{array}\]]]> where in the last step we use the inequality a1+a2+⋯+anb1+b2+⋯+bn⩽maxa1b1,a2b2,…,anbn, provided that n⩾1 and ai,bi>0 0$]]> for i∈{1,2,…,n} .

Since FSn∈OL for all n⩾κ , we get from (11) that lim supx→∞P(Sη>x−1)P(Sη>x)<∞, x-1)}{\mathbb{P}(S_{\eta }>x)}<\infty ,\]]]> 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 P(Sη>x)= ∑n=1∞P(Sn>x)P(η=n)⩾P(Sκ>x)P(η=κ)⩾F‾ξκ(x)P(η=κ). x)& \displaystyle ={\sum \limits_{n=1}^{\infty }}\mathbf{P}(S_{n}>x)\mathbb{P}(\eta =n)\\{} & \displaystyle \geqslant \mathbb{P}(S_{\kappa }>x)\mathbb{P}(\eta =\kappa )\\{} & \displaystyle \geqslant \overline{F}_{\xi _{\kappa }}(x)\mathbb{P}(\eta =\kappa ).\end{array}\]]]> Similarly, for K⩾2 and x⩾2K , P(Sη>x−1)= ∑n=1κP(Sn>x−1)P(η=n)+∑1⩽k⩽(x−1)/(K−1)P(Sκ+k>x−1)P(η=κ+k)+∑k>(x−1)/(K−1)P(x−1<Sκ+k⩽x)P(η=κ+k)+∑k>(x−1)/(K−1)P(Sκ+k>x)P(η=κ+k):=K1(x)+K2(x)+K3(x)+K4(x). x-1)& \displaystyle ={\sum \limits_{n=1}^{\kappa }}\mathbf{P}(S_{n}>x-1)\mathbb{P}(\eta =n)\\{} & \displaystyle \hspace{1em}+\sum \limits_{1\leqslant k\leqslant (x-1)/(K-1)}\mathbf{P}(S_{\kappa +k}>x-1)\mathbb{P}(\eta =\kappa +k)\\{} \hspace{2.5pt}& \displaystyle \hspace{1em}+\sum \limits_{k>(x-1)/(K-1)}\mathbf{P}(x-1(x-1)/(K-1)}\mathbf{P}(S_{\kappa +k}>x)\mathbb{P}(\eta =\kappa +k)\\{} \hspace{2.5pt}& \displaystyle :=\mathcal{K}_{1}(x)+\mathcal{K}_{2}(x)+\mathcal{K}_{3}(x)+\mathcal{K}_{4}(x).\end{array}\]]]> The distribution function FSκ belongs to the class OL due to Lemma 4. So, by estimate (6) we have lim supx→∞K1(x)P(Sη>x)=lim supx→∞J1(x)<∞. x)}=\underset{x\to \infty }{\limsup }\mathcal{J}_{1}(x)<\infty .\]]]> Now we consider the sum K2(x) . Since FSκ is O -exponential, we have supx⩾0P(Sκ>x−1)P(Sκ>x)⩽c4 x-1)}{\mathbb{P}(S_{\kappa }>x)}\leqslant c_{4}\]]]> with some positive constant c4 . On the other hand, the third condition of Theorem 6 implies that supx⩾c5F‾ξκ+k(x−1)F‾ξκ+k(x)⩽c6 for some constants c5>2 2$]]>, c6>0 0$]]> and all k∈N .

By Lemma 1 (with v=c5 ) we have P(Sκ+1>x−1)P(Sκ+1>x)⩽max{supz⩾x−c5+1P(Sκ>z−1)P(Sκ>z),supz⩾c5F‾ξκ+1(z−1)F‾ξκ+1(z)}. x-1)}{\mathbb{P}(S_{\kappa +1}>x)}\leqslant \max \bigg\{\underset{z\geqslant x-c_{5}+1}{\sup }\frac{\mathbb{P}(S_{\kappa }>z-1)}{\mathbb{P}(S_{\kappa }>z)},\underset{z\geqslant c_{5}}{\sup }\frac{\overline{F}_{\xi _{\kappa +1}}(z-1)}{\overline{F}_{\xi _{\kappa +1}}(z)}\bigg\}.\]]]> Consequently, supx⩾c5P(Sκ+1>x−1)P(Sκ+1>x)⩽maxc4,c6:=c7. x-1)}{\mathbb{P}(S_{\kappa +1}>x)}\leqslant \max \left\{c_{4},c_{6}\right\}:=c_{7}.\]]]> Applying Lemma 1 again for the sum Sκ+2=Sκ+1+ξκ+2 (with v=x/2+1/2 ), we get P(Sκ+2>x−1)P(Sκ+2>x)⩽max{supz⩾x2+12P(Sκ+1>z−1)P(Sκ+1>z),supz⩾x2+12F‾ξκ+2(z−1)F‾ξκ+2(z)}. x-1)}{\mathbb{P}(S_{\kappa +2}>x)}\leqslant \max \bigg\{\underset{z\geqslant \frac{x}{2}+\frac{1}{2}}{\sup }\frac{\mathbb{P}(S_{\kappa +1}>z-1)}{\mathbb{P}(S_{\kappa +1}>z)},\underset{z\geqslant \frac{x}{2}+\frac{1}{2}}{\sup }\frac{\overline{F}_{\xi _{\kappa +2}}(z-1)}{\overline{F}_{\xi _{\kappa +2}}(z)}\bigg\}.\]]]> If x⩾2(c5−1)+1 , then x/2+1/2⩾c5 . Therefore, by the last inequality we obtain that supx⩾2(c5−1)+1P(Sκ+2>x−1)P(Sκ+2>x)⩽c7. x-1)}{\mathbb{P}(S_{\kappa +2}>x)}\leqslant c_{7}.\]]]> Applying Lemma 1 once again (with v=x/3+2/3 ), we get P(Sκ+3>x−1)P(Sκ+3>x)⩽max{supz⩾2x3+13P(Sκ+2>z−1)P(Sκ+2>z),supz⩾x3+23F‾ξκ+3(z−1)F‾ξκ+3(z)}. x-1)}{\mathbb{P}(S_{\kappa +3}>x)}\leqslant \max \bigg\{\underset{z\geqslant \frac{2x}{3}+\frac{1}{3}}{\sup }\frac{\mathbb{P}(S_{\kappa +2}>z-1)}{\mathbb{P}(S_{\kappa +2}>z)},\underset{z\geqslant \frac{x}{3}+\frac{2}{3}}{\sup }\frac{\overline{F}_{\xi _{\kappa +3}}(z-1)}{\overline{F}_{\xi _{\kappa +3}}(z)}\bigg\}.\]]]> If x⩾3(c5−1)+1 , then 2x/3+1/3⩾2(c5−1)+1 and x/3+2/3⩾c5 . So, the last estimate implies supx⩾3(c5−1)+1P(Sκ+3>x−1)P(Sκ+3>x)⩽c7. x-1)}{\mathbb{P}(S_{\kappa +3}>x)}\leqslant c_{7}.\]]]> Continuing the process, we can get that supx⩾k(c5−1)+1P(Sκ+k>x−1)P(Sκ+k>x)⩽c7 x-1)}{\mathbb{P}(S_{\kappa +k}>x)}\leqslant c_{7}\]]]> for all k∈N .

We can suppose that K=c5 in representation (14). In such a case, it follows from inequality (16) that lim supx→∞K2(x)P(Sη>x)⩽lim supx→∞c7P(Sη>x)∑1⩽k⩽x−1c5−1P(Sκ+k>x)P(η=κ+k)⩽c7. x)}& \displaystyle \leqslant \underset{x\to \infty }{\limsup }\frac{c_{7}}{\mathbb{P}(S_{\eta }>x)}\sum \limits_{1\leqslant k\leqslant \frac{x-1}{c_{5}-1}}\mathbb{P}(S_{\kappa +k}>x)\mathbb{P}(\eta =\kappa +k)\\{} & \displaystyle \leqslant c_{7}.\end{array}\]]]> Since, obviously, lim supx→∞K4(x)P(Sη>x)⩽1, x)}\leqslant 1,\]]]> it remains to estimate sum K3(x) . Using Lemma 2, we obtain K3(x)⩽A∑k>x−1c5−1P(η=κ+k)( ∑l=1k(1−supx∈RP(x−1⩽ξκ+l⩽x)))−1/2 \frac{x-1}{c_{5}-1}}\mathbb{P}(\eta =\kappa +k)\Bigg({\sum \limits_{l=1}^{k}}{\Big(1-\underset{x\in \mathbb{R}}{\sup }\mathbb{P}(x-1\leqslant \xi _{\kappa +l}\leqslant x)\Big)\Bigg)}^{-1/2}\]]]> with some absolute positive constant A. By the fourth condition of the theorem, 1k ∑l=1ksupx∈R(F‾ξκ+l(x−1)−F‾ξκ+l(x))⩽1−Δ for some 0<Δ<1 and all sufficiently large k. So, for such k, ∑l=1k(1−supx∈RP(x−1⩽ξκ+l⩽x))⩾kΔ. From the last estimate it follows that K3(x)⩽AΔ∑k>x−1c5−11kP(η=κ+k)⩽AΔc5−1x−1P(η>κ+x−1c5−1) \frac{x-1}{c_{5}-1}}\frac{1}{\sqrt{k}}\mathbb{P}(\eta =\kappa +k)\\{} & \displaystyle \leqslant \frac{A}{\sqrt{\Delta }}\sqrt{\frac{c_{5}-1}{x-1}}\mathbb{P}\bigg(\eta >\kappa +\frac{x-1}{c_{5}-1}\bigg)\end{array}\]]]> for sufficiently large x. Therefore, lim supx→∞K3(x)P(Sη>x)⩽AΔc5−1P(η=κ)lim supx→∞F‾η(x−1c5−1)x−1F‾ξκ(x−1)lim supx→∞F‾ξκ(x−1)F‾ξκ(x)<∞ x)}\\{} & \displaystyle \hspace{1em}\leqslant \frac{A}{\sqrt{\Delta }}\frac{\sqrt{c_{5}-1}}{\mathbb{P}(\eta =\kappa )}\underset{x\to \infty }{\limsup }\frac{\overline{F}_{\eta }(\frac{x-1}{c_{5}-1})}{\sqrt{x-1}\hspace{2.5pt}\overline{F}_{\xi _{\kappa }}(x-1)}\underset{x\to \infty }{\limsup }\frac{\overline{F}_{\xi _{\kappa }}(x-1)}{\overline{F}_{\xi _{\kappa }}(x)}\\{} & \displaystyle \hspace{1em}<\infty \end{array}\]]]> by estimate (13) and the last condition of the theorem. Representation (14) and estimates (15), (17), (18), and (19) imply the desired inequality (12). Theorem 6 is proved. □

 Examples of <inline-formula id="j_vmsta52_ineq_221"><alternatives> <mml:math><mml:mi mathvariant="script">O</mml:mi></mml:math> <tex-math><![CDATA[$\mathcal{O}$]]></tex-math></alternatives></inline-formula>-exponential random sums

In this section, we present three examples of random sums Sη for which the d.f.s FSη are O -exponential.

Let {ξ1,ξ2,…} be independent r.v.s. We suppose that the r.v. ξk for k∈{1,2,…,D} is distributed according to the Pareto law with parameters k and α, that is, F‾ξk(x)=kk+xα,x⩾0, where k∈{1,2,…,D} , D⩾1 , and α>0 0$]]>. In addition, we suppose that the r.v. ξD+k for each k∈N is distributed according to the exponential law with parameter λ/k , that is, F‾ξD+k(x)=e−λx/k,x⩾0.

It follows from Theorem 4 that the d.f. of the random sum Sη is O -exponential for each counting r.v. η independent of {ξ1,ξ2,…} under the condition P(η=κ)>0 0$]]> for some κ∈{1,2,…,D} because:

Fξk∈L⊂OLfor eachk⩽κ ,

supx⩾0supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)=max{sup0⩽x⩽1supk⩾11F‾ξκ+k(x),supx>1supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)}=max{sup0⩽x⩽1max{max1⩽k⩽D−κ(κ+k+xκ+k)α,supk⩾1eλx/k},supx>1max{max1⩽k⩽D−κ(κ+k+xκ+k+x−1)α,supk⩾1eλ/k}}⩽max{2α,eλ}. 1}{\sup }\underset{k\geqslant 1}{\sup }\frac{\overline{F}_{\xi _{\kappa +k}}(x-1)}{\overline{F}_{\xi _{\kappa +k}}(x)}\bigg\}\\{} & \displaystyle \hspace{1em}=\max \bigg\{\underset{0\leqslant x\leqslant 1}{\sup }\max \bigg\{\underset{1\leqslant k\leqslant D-\kappa }{\max }{\bigg(\frac{\kappa +k+x}{\kappa +k}\bigg)}^{\alpha },\hspace{2.5pt}\underset{k\geqslant 1}{\sup }\hspace{0.1667em}{\mathrm{e}}^{\lambda x/k}\bigg\},\\{} & \displaystyle \hspace{2em}\underset{x>1}{\sup }\max \bigg\{\underset{1\leqslant k\leqslant D-\kappa }{\max }{\bigg(\frac{\kappa +k+x}{\kappa +k+x-1}\bigg)}^{\alpha },\hspace{2.5pt}\underset{k\geqslant 1}{\sup }\hspace{0.1667em}{\mathrm{e}}^{\lambda /k}\bigg\}\bigg\}\\{} & \displaystyle \hspace{1em}\leqslant \max \big\{{2}^{\alpha },\hspace{0.1667em}{\mathrm{e}}^{\lambda }\big\}.\end{array}$]]>

Let a r.v. η be uniformly distributed on {1,2,…,D} , that is, P(η=k)=1D,k∈{1,2,…,D}, for some D⩾2 . Let {ξ1,ξ2,…,ξD} be independent r.v.s, where ξ1 is exponentially distributed, and ξ2,…,ξD are uniformly distributed.

If the r.v. η is independent of the r.v.s {ξ1,ξ2,…,ξD} , then Theorem 5 implies that the d.f. of the random sum Sη is O -exponential.

Let {ξ1,ξ2,…} be independent r.v.s, where {ξ1,ξ2,…,ξκ−1} are finitely supported, κ⩾2 , and ξκ is distributed according to the Weibull law, that is, F‾ξκ(x)=e−x,x⩾0. In addition, we suppose that the r.v. ξκ+k for each k=m2 , m⩾2 , has the d.f. with tail F‾ξκ+k(x)=1ifx<0,1kif0⩽x<k,1ke−(x−k)ifx⩾k, whereas for each remaining index k∉{m2,m∈N∖{1}} , the r.v. ξκ+k has the exponential distribution, that is, F‾ξκ+k(x)=e−x,x⩾0.

If the counting r.v. η is independent of {ξ1,ξ2,…} and is distributed according to the Poisson law with parameter λ, then it follows from Theorem 6 that the random sum Sη is O -exponentially distributed because:

Fξκ∈L⊂OL ;

limx→∞F‾ξk(x)F‾ξκ(x)=0ifk=1,2,…,κ−1 ;

supx⩾1supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)=supx⩾1max{supk⩾1,k=m2,m⩾2F‾ξκ+k(x−1)F‾ξκ+k(x),supk⩾1,k≠m2F‾ξκ+k(x−1)F‾ξκ+k(x)}=supx⩾1max{supk⩾1,k=m2,m⩾2{1[1,k)(x)+ex−k1[k,k+1)(x)+e1[k+1,∞)(x)},supk⩾1,k≠m2e}=e;

lim supk→∞1k ∑l=1ksupx⩾0(F‾ξκ+l(x−1)−F‾ξκ+l(x))=lim supk→∞1k( ∑l=1,l=m2k(1−1l)+(1−1e) ∑l=1,l≠m2k1)⩽(1−1e);

F‾η(x)<(eλx)x,x>λ \lambda $]]>.

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 {ξ1,ξ2,…} from the last example satisfy the conditions of Theorem 6, whereas the third condition of Theorem 4 does not hold because, in this case, supx⩾0supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)⩾sup0⩽x<1supk⩾1F‾ξκ+k(x−1)F‾ξκ+k(x)⩾sup0⩽x<1supk=m2,m⩾2k=∞.

Acknowledgments

We would like to thank the anonymous referees for the detailed and helpful comments on the first and second versions of the manuscript.

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