VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA74 10.15559/17-VMSTA74 Research Article Randomly stopped maximum and maximum of sums with consistently varying distributions AndrulytėIeva Marijai.m.andrulyte@gmail.com ManstavičiusMartynasmartynas.manstavicius@mif.vu.lt ŠiaulysJonasjonas.siaulys@mif.vu.lt∗ Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania Corresponding author. 2017 632017416578 10122016 2312017 2712017 © 2017 The Author(s). Published by VTeX2017 Open access article under the CC BY license.

Let {ξ1,ξ2,…} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let S0:=0 and Sn:=ξ1+ξ2+⋯+ξn for n⩾1 . We consider conditions for random variables {ξ1,ξ2,…} and η under which the distribution functions of the random maximum ξ(η):=max{0,ξ1,ξ2,…,ξη} and of the random maximum of sums S(η):=max{S0,S1,S2,…,Sη} belong to the class of consistently varying distributions. In our consideration the random variables {ξ1,ξ2,…} are not necessarily identically distributed.

 Heavy tail consistently varying tail randomly stopped maximum randomly stopped maximum of sums 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 , Sn=ξ1+ξ2+⋯+ξn for n∈N , and let Sη= ∑k=1ηξk be the randomly stopped sum of r.v.s {ξ1,ξ2,…} .

Similarly, let ξ(n)=max{0,ξ1,ξ2,…,ξn} for n∈N , ξ(0)=0 , and let ξ(η)=max{0,ξ1,ξ2,…,ξη}ifη⩾1,0ifη=0 be the randomly stopped maximum of r.v.s {ξ1,ξ2,…} .

Finally, let S(n):=max{S0,S1,S2,…,Sn} for n⩾0 , and let S(η):=max{S0,S1,S2,…,Sη} be the randomly stopped maximum of sums S0,S1,S2,… .

We denote the distribution functions (d.f.s) of Sη , ξ(η) , and S(η) by FSη , Fξ(η) , and FS(η) respectively, together with their tails F‾Sη , F‾ξ(η) , and F‾S(η) . For any positive x, we have F‾Sη(x)= ∑n=1∞P(η=n)P(Sn>x),F‾ξ(η)(x)= ∑n=1∞P(η=n)P(ξ(n)>x),F‾S(η)(x)= ∑n=1∞P(η=n)P(S(n)>x). x),\\{} \displaystyle \overline{F}_{\xi _{(\eta )}}(x)& \displaystyle ={\sum \limits_{n=1}^{\infty }}\mathbb{P}(\eta =n)\mathbb{P}(\xi _{(n)}>x),\\{} \displaystyle \overline{F}_{S_{(\eta )}}(x)& \displaystyle ={\sum \limits_{n=1}^{\infty }}\mathbb{P}(\eta =n)\mathbb{P}(S_{(n)}>x).\end{array}\]]]>

In [14], conditions were found for the d.f.s FSη to belong to the class of consistently varying distributions. In this paper, we are interested in sufficient conditions under which the d.f.s Fξ(η) and FS(η) have consistently varying tails.

Throughout this paper, for two vanishing (at infinity) functions f and g, f(x)=o(g(x)) means that limx→∞f(x)/g(x)=0 , and f(x)∼g(x) means that limx→∞f(x)/g(x)=1 . Also, we denote the support of a counting r.v. η by supp(η):={n∈Z+:P(η=n)>0}. 0\big\}.\]]]>

Before discussing the properties of Fξ(η) and FS(η) , we recall the definitions of some classes of heavy-tailed d.f.s. For a d.f. F, we denote F‾(x)=1−F(x) for real x.

A d.f. F is heavy-tailed (F∈H) if for every δ>0 0$]]>, limx→∞F‾(x)eδx=∞.

A d.f. F is long-tailed (F∈L) if for every y (equivalently, for some y>0 0$]]>), F‾(x+y)∼F‾(x) .

A d.f. F has dominatingly varying tail (F∈D) if for every y∈(0,1) (equivalently, for some y∈(0,1) ), lim supx→∞F‾(xy)F‾(x)<∞.

A d.f. F has consistently varying tail (F∈C) if limy↑1lim supx→∞F‾(xy)F‾(x)=1.

A d.f. F has regularly varying tail (F∈R) if there is α⩾0 such that limx→∞F‾(xy)F‾(x)=y−α, for all y>0 0$]]>.

A d.f. F supported on the interval [0,∞) is subexponential (F∈S) if limx→∞F∗F‾(x)F‾(x)=2. If a d.f. G is supported on R , then we say that G is subexponential (G∈S) if the d.f. F(x)=G(x)1[0,∞)(x) satisfies relation (1).

It is known (see, e.g., [5, 12, 15], and Chapters 1.4 and A3 in [10]) that these classes satisfy the following inclusions: R⊂C⊂L∩D⊂S⊂L⊂H,D⊂H. These inclusions are depicted in Fig. 1 borrowed from the paper [14]. In this figure, the class C of distributions having consistently varying tails is highlighted.

Classes of heavy-tailed distributions

It should be noted that the subject of the paper is partially motivated by the closure problem of the random convolution. In the case of independent and identically distributed (i.i.d.) r.v.s {ξ,ξ1,ξ2,…} , we say that a class K of d.f.s is closed with respect to the random convolution if the condition Fξ∈K implies FSη∈K . The first result on the convolution closure of subexponential distributions was obtained by Embrechts and Goldie (see Thm. 4.2 in [11]) and by Cline (see Thm. 2.13 in [6]).

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 the d.f. FSη∈S .

The random closure results for class D can be found in [7, 16], and for the class L , in [1, 16, 19, 20]. The random closure results for the class C can be derived from the results of [14]. We note that in [7, 14, 20], the case of not necessarily identically distributed r.v.s {ξ1,ξ2,…} was considered. We further recall two known results. First, in Theorem 2, we give conditions for  FSη to belong to the class D . Its proof can be found in [7, Thm. 2.1]. Recall that a d.f. F belongs to the class D if and only if its upper Matuszewska index JF+<∞ , where by definition JF+=−limy→∞1logylog(lim infx→∞F‾(xy)F‾(x)).

Let r.v.s {ξ1,ξ2,…} be nonnegative independent but not necessary identically distributed, and η be a counting r.v. independent of {ξ1,ξ2,…} . Then the d.f.  FSη belongs to the class D if the following three conditions are satisfied: (i)Fξκ∈Dforsomeκ∈supp(η),(ii)lim supx→∞supn⩾κ1nF‾ξκ(x) ∑i=1nF‾ξi(x)<∞,(iii)Eηp+1<∞forsomep>JFξκ+. {J_{F_{\xi _{\kappa }}}^{+}}.\end{array}\]]]>

A similar result for r.v.s having d.f.s with consistently varying tails can be found in [14, Thm. 6].

Let {ξ1,ξ2,…} be independent real-valued r.v.s, and η be a counting r.v. independent of {ξ1,ξ2,…} . Then the d.f. FSη belongs to the class C if the following conditions are satisfied: (a)Fξ1∈C,(b)foreachk⩾2,eitherFξk∈CorF‾ξk(x)=o(F‾ξ1(x)),(c)lim supx→∞supn⩾11nF‾ξ1(x) ∑i=1nF‾ξi(x)<∞,(d)Eηp+1<∞forsomep>JFξ1+. {J_{F_{\xi _{1}}}^{+}}.\end{array}\]]]>

In this work, we consider the randomly stopped maximum ξ(η) and randomly stopped maximum of sums S(η) of independent but not necessarily identically distributed r.v.s. As was noted before, we restrict our consideration to the class C but extend it to the real-valued r.v.s as in Theorem 3.

If r.v.s {ξ1,ξ2,…} are not identically distributed, then different collections of conditions on r.v.s {ξ1,ξ2,…} and η imply that Fξ(η)∈C or FS(η)∈C . We suppose that some r.v.s from {ξ1,ξ2,…} have distributions belonging to the class C , and we find conditions for r.v.s {ξ1,ξ2,…} and η such that the distribution of the randomly stopped maximum or the randomly stopped maximum of sums remains in the same class. The results presented and their proofs are closely related to the results of the papers [7, 8, 14].

It is worth noting that the closure properties for d.f.s FS(η) in the case of i.i.d. r.v.s can be derived, for instance, from the asymptotic formulas obtained in [9, 13, 17, 18, 21]. Unfortunately, in the case of nonidentically distributed r.v.s, similar asymptotic formulas do not exist. Therefore we have to use other methods to prove our main results.

The rest of the paper is organized as follows. In Section 2, we present our main results together with a few examples of randomly stopped maximum ξ(η) and randomly stopped maximum of sums S(η) 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.

 Main results

In this section, we present four statements, two theorems and two corollaries. Theorem 4 and Corollary 1 deal with the belonging of the d.f. Fξ(η) to the class  C .

Let {ξ1,ξ2,…} be a sequence of independent real-valued r.v.s, and let η be a counting r.v. independent of {ξ1,ξ2,…} . The d.f. of the randomly stopped maximum Fξ(η) belongs to the class C if the following conditions hold: (a)Fξϰ∈Cfor someϰ∈supp(η),(b)for eachk≠ϰ,eitherFξk∈CorF‾ξk(x)=o(F‾ξϰ(x)),(c)lim supx→∞supn⩾11φ(n)F‾ξϰ(x) ∑k=1nF‾ξk(x)<∞, where {φ(n)}n=1∞ is a positive sequence such that E(φ(η)1[1,∞)(η))<∞ .

If r.v.s {ξ1,ξ2,…} are identically distributed with common d.f. Fξ∈C , then conditions (a), (b), and (c) are satisfied with φ(n)=n for n∈N . Hence, the following statement immediately follows from Theorem 4.

Let {ξ1,ξ2,…} be a sequence of i.i.d. real-valued r.v.s with common d.f. Fξ∈C , and let η be a counting r.v. independent of {ξ1,ξ2,…} . The d.f. of the randomly stopped maximum Fξ(η) belongs to the class C if   Eη   is finite.

In Theorem 5 and Corollary 2, we present conditions under which the d.f. of the randomly stopped maximum of sums S(η) has consistently varying tail.

Let {ξ1,ξ2,…} be independent real-valued r.v.s, and let η be a counting r.v. independent of {ξ1,ξ2,…} . The d.f. FS(η) belongs to the class C if the following conditions hold: (a)Fξk∈Cfor eachk∈N,(b)lim supx→∞supn⩾11nF‾ξ1(x) ∑k=1nF‾ξk(x)<∞,(c)Eηp+1<∞for somep>JFξ1+. {J_{F_{\xi _{1}}}^{+}}.\end{array}\]]]>

Similarly to Corollary 1, we can state the following corollary. We note that in the i.i.d. case condition (b) is obviously satisfied if F‾ξ1(x)>0 0$]]> for all x∈R .

Let {ξ1,ξ2,…} be i.i.d. real-valued r.v.s with common d.f. Fξ∈C , and let η is a counting r.v. independent of {ξ1,ξ2,…} . The d.f.  FS(η) belongs to the class C if Eηp+1<∞ for some p>JFξ+ {J_{F_{\xi }}^{+}}$]]>.

Further in this section, we present two examples of r.v.s {ξ1,ξ2,…} and η showing the applicability of our theorems.

Let {ξ1,ξ2,…} be independent r.v.s such that:

r.v.s ξk are exponentially distributed for k≡0mod3 , that is, F‾ξk(x)=1(−∞,0)(x)+e−x1[0,∞)(x),k∈{3,6,9,…};

r.v.s ξk are degenerate at zero for all k≡2mod3 and k≡1mod3,k⩾4 , that is, Fξk(x)=1[0,∞)(x),k∈{2,5,8,…}∪{4,7,10,…};

ξ1=(1+U)2G , where r.v.s U and G are independent, U is uniformly distributed on the interval [0,1] , and G is geometrically distributed with parameter q∈(0,1) , that is, P(G=l)=(1−q)ql,l∈{0,1,2,…}. In addition, let η be a counting r.v. independent of {ξ1,ξ2,…} .

Theorem 4 implies that the d.f. of the randomly stopped maximum ξ(η) belongs to the class C if 1∈supp(η) and Eη<∞ because:

Fξ1∈C , but Fξ1∉R due to considerations in pp. 122–123 of [2],

F‾ξk(x)=o(F‾ξ1(x)) for each k≠1 ,

supn⩾11nF‾ξ1(x) ∑k=1nF‾ξk(x)⩽2 for sufficiently large x.

Let {ξ1,ξ2,…} be independent r.v.s distributed according to Pareto-type laws, namely, F‾ξk(x)=1(−∞,0)(x)+1(1+x)31[0,∞)(x)ifk∈{1,3,5,…},F‾ξk(x)=1(−∞,−1)(x)+1(2+x)31[−1,∞)(x)ifk∈{2,4,6,…}. Further, let η be a counting r.v. independent of {ξ1,ξ2,…} that has the zeta distribution with parameter 6, that is, P(η=m)=1ζ(6)1(m+1)6,m∈{0,1,2,…}, where ζ denotes the Riemann zeta function.

Theorem 5 implies that the d.f. of the randomly stopped maximum of sums S(η) belongs to the class C because:

Fξk∈R⊂C for each k∈N ;

supn⩾11nF‾ξ1(x) ∑k=1nF‾ξk(x)=1 for positive x;

JFξ1+=3 and Eη4.5<∞ .

Auxiliary lemmas

In this section, we give auxiliary lemmas, which we use in the proof of Theorem 5. The first lemma was proved in [17, Thm. 2.1]; a more general case can be found in [3, Thm. 2.1]. We recall only that C⊂L . Hence, the statement of the next lemma holds for d.f.s with consistently varying tails.

Let {X1,X2,…,Xn} be independent real-valued r.v.s such that FXk∈L for all k∈N . Then P(max1⩽k⩽n ∑i=1kXi>x)∼x→∞P( ∑i=1nXi>x). x\Bigg)\underset{x\to \infty }{\sim }\mathbb{P}\Bigg({\sum \limits_{i=1}^{n}}X_{i}>x\Bigg).\]]]>

The second auxiliary lemma was proved in [14, Lemma 3]. It describes the situation where the d.f. of sums of independent r.v.s belongs to the class C .

Let {X1,X2,…,Xn} be independent real-valued r.v.s. The d.f. of the sum Σn:=X1+X2+⋯+Xn belongs to the class C if the following two conditions are satisfied: (a)FX1∈C,(b)for eachk∈{2,…,n},eitherFXk∈CorF‾Xk(x)=o(F‾X1(x)).

The following statement was proved in [7, Lemma 3.2]. It gives an upper estimate for the tail of the sum of r.v.s from the class D .

Let {X1,X2,…} be nonnegative independent r.v.s, FXν∈D for some ν⩾1 such that lim supx→∞supn⩾ν1nF‾Xν(x) ∑i=1nF‾Xi(x)<∞. Then, for each p>JFXν+ {J_{F_{X_{\nu }}}^{+}}$]]>, there exists a positive constant c1 such that F‾Σn(x)⩽c1np+1F‾Xν(x), for all n⩾ν and x⩾0 , where FΣn is the d.f. of the sum Σn=X1+⋯+Xn .

The last useful lemma is an obvious conclusion from Theorem 3.1 in [4].

Let {X1,X2,…Xn} be independent real-valued r.v.s. If FXk∈C for each k∈{1,2,…,n} , then P( ∑i=1nXi>x)∼ ∑i=1nF‾Xi(x). x\Bigg)\sim {\sum \limits_{i=1}^{n}}\overline{F}_{X_{i}}(x).\]]]>

Proofs of the main results

It suffices to prove that lim supy↑1lim supx→∞F‾ξ(η)(xy)F‾ξ(η)(x)⩽1. For all x>0 0$]]> and K∈N , we have F‾ξ(η)(x)= ∑n=1∞F‾ξ(n)(x)P(η=n)=( ∑n=1K+ ∑n=K+1∞)P( ⋃k=1n{ξk>x})P(η=n). x\}\Bigg)\mathbb{P}(\eta =n).\end{array}\]]]> Therefore, F‾ξ(η)(xy)F‾ξ(η)(x)=∑n=1KP(⋃k=1n{ξk>xy})P(η=n)P(ξ(η)>x)+∑n=K+1∞P(⋃k=1n{ξk>xy})P(η=n)P(ξ(η)>x)=:J1+J2, xy\}\big)\mathbb{P}(\eta =n)}{\mathbb{P}(\xi _{(\eta )}>x)}\\{} & \displaystyle \hspace{1em}+\frac{{\textstyle\sum _{n=K+1}^{\infty }}\mathbb{P}\big({\textstyle\bigcup _{k=1}^{n}}\{\xi _{k}>xy\}\big)\mathbb{P}(\eta =n)}{\mathbb{P}(\xi _{(\eta )}>x)}\\{} & \displaystyle =:\mathcal{J}_{1}+\mathcal{J}_{2},\end{array}\]]]> for all x>0 0$]]> and y∈(0,1) .

Denote K:={k∈N:Fξk∉CandF‾ξk(x)=o(F‾ξϰ(x))}.

If x>0 0$]]>, 1/2⩽y<1 , and K⩾ϰ , then J1⩽1F‾ξ(η)(x) ∑n=1K ∑k=1k∉KnF‾ξk(xy)P(η=n)+ ∑n=1K ∑k=1k∈KnF‾ξk(x/2)F‾ξ(η)(x)P(η=n).

If k∈K , then F‾ξk(x/2)=o(F‾ξ(η)(x)), because F‾ξk(x/2)F‾ξ(η)(x)⩽F‾ξk(x/2)F‾ξ(ϰ)(x)P(η=ϰ)⩽F‾ξk(x/2)F‾ξϰ(x/2)P(η=ϰ)F‾ξϰ(x/2)F‾ξϰ(x), and Fξκ∈C⊂D .

The obtained asymptotic relation (5) and estimate (4) imply lim supx→∞J1⩽lim supx→∞1F‾ξ(η)(x) ∑n=1K ∑k=1k∉KnF‾ξk(xy)P(η=n)⩽lim supx→∞(max1⩽k⩽Kk∉K{F‾ξk(xy)F‾ξk(x)}1F‾ξ(η)(x) ∑n=1K ∑k=1k∉KnF‾ξk(x)P(η=n))⩽max1⩽k⩽Kk∉K{lim supx→∞F‾ξk(xy)F‾ξk(x)}×lim supx→∞{1F‾ξ(η)(x) ∑n=1K ∑k=1nF‾ξk(x)P(η=n)}.

For each 1⩽n⩽K , we have F‾ξ(n)(x)=P( ⋃k=1n{ξk>x})⩾ ∑k=1nF‾ξk(x)(1− ∑k=1KF‾ξk(x)), x\}\Bigg)\geqslant {\sum \limits_{k=1}^{n}}\overline{F}_{\xi _{k}}(x)\Bigg(1-{\sum \limits_{k=1}^{K}}\overline{F}_{\xi _{k}}(x)\Bigg),\]]]> due to the Bonferroni inequality. This implies that, for an arbitrary ε>0 0$]]>, ∑k=1nF‾ξk(x)⩽(1+ε)F‾ξ(n)(x), if x is sufficiently large.

Substituting the last estimate into inequality (6), we get lim supy↑1lim supx→∞J1⩽(1+ε)max1⩽k⩽Kk∉K{lim supy↑1lim supx→∞F‾ξk(xy)F‾ξk(x)}=1+ε, because Fξk∈C for each k∉K .

Since ε>0 0$]]> is arbitrary, the last estimate implies that lim supy↑1lim supx→∞J1⩽1.

Since P(ξ(η)>x)⩾P(ξ(ϰ)>x)P(η=ϰ)⩾P(ξϰ>x)P(η=ϰ), x)\geqslant \mathbb{P}(\xi _{(\varkappa )}>x)\mathbb{P}(\eta =\varkappa )\geqslant \mathbb{P}(\xi _{\varkappa }>x)\mathbb{P}(\eta =\varkappa ),\]]]> we obtain J2⩽∑n=K+1∞∑k=1nF‾ξk(xy)P(η=n)F‾ξϰ(x)P(η=ϰ), for all x>0 0$]]> and y∈(0,1) .

Condition (c) of the theorem implies that, for some constant c2>0 0$]]>, ∑k=1nF‾ξk(x)⩽c2φ(n)F‾ξϰ(x), for all sufficiently large x and all n⩾1 .

Consequently, lim supy↑1lim supx→∞J2⩽lim supy↑1lim supx→∞c2F‾ξϰ(xy)∑n=K+1∞φ(n)P(η=n)F‾ξϰ(x)P(η=ϰ)=c2P(η=ϰ)(lim supy↑1lim supx→∞F‾ξϰ(xy)F‾ξϰ(x)) ∑n=K+1∞φ(n)P(η=n).

Relations (3), (7), and (8) imply that lim supy↑1lim supx→∞F‾ξ(η)(xy)F‾ξ(η)(x)⩽1+c2P(η=ϰ)E(φ(η)1[K+1,∞)(η)), for arbitrary K⩾ϰ .

The desired inequality (2) now follows from the last estimate because E(φ(η)1[1,∞)(η)) is finite due to the conditions of the theorem. Theorem 4 is proved.  □

Similarly as in the proof of Theorem 4, it suffices to show that lim supy↑1lim supx→∞F‾S(η)(xy)F‾S(η)(x)⩽1.

If K∈N and x>0 0$]]>, then P(S(η)>x)=( ∑n=1K+ ∑n=K+1∞)P(S(n)>x)P(η=n). x)=\Bigg({\sum \limits_{n=1}^{K}}+{\sum \limits_{n=K+1}^{\infty }}\Bigg)\mathbb{P}(S_{(n)}>x)\mathbb{P}(\eta =n).\]]]>

Therefore, P(S(η)>xy)P(S(η)>x)=∑n=1KP(S(n)>xy)P(η=n)P(S(η)>x)+∑n=K+1∞P(S(n)>xy)P(η=n)P(S(η)>x)=:I1+I2, \mathit{xy})}{\mathbb{P}(S_{(\eta )}>x)}& \displaystyle =\frac{{\textstyle\sum _{n=1}^{K}}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta =n)}{\mathbb{P}(S_{(\eta )}>x)}\\{} & \displaystyle \hspace{1em}+\frac{{\textstyle\sum _{n=K+1}^{\infty }}\mathbb{P}(S_{(n)}>\mathit{xy})\mathbb{P}(\eta =n)}{\mathbb{P}(S_{(\eta )}>x)}\\{} & \displaystyle =:\mathcal{I}_{1}+\mathcal{I}_{2},\end{array}\]]]> for all x>0 0$]]> and y∈(0,1) .

The r.v. η is not degenerate at zero. Therefore, there exists a∈N such that P(η=a)>0 0$]]>. If K⩾a , then I1⩽∑n=1KP(S(n)>xy)P(η=n)∑n=1KP(S(n)>x)P(η=n)⩽max1⩽n⩽Kn∈supp(η)P(S(n)>xy)P(S(n)>x), \mathit{xy})\mathbb{P}(\eta =n)}{{\textstyle\sum _{\displaystyle n=1}^{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)}>\mathit{xy})}{\mathbb{P}(S_{(n)}>x)},\]]]> due to the inequality a1+a2+⋯+amb1+b2+⋯+bm⩽max{a1b1,a2b2,…,ambm}, provided for all ai⩾0 , bi>0 0$]]>, i∈{1,2,…,m} , and m∈N .

Since C⊂L , using Lemma 1, we obtain lim supy↑1lim supx→∞I1⩽lim supy↑1lim supx→∞max1⩽n⩽Kn∈supp(η)F‾S(n)(xy)F‾S(n)(x)=max1⩽n⩽Kn∈supp(η)lim supy↑1lim supx→∞F‾S(n)(xy)F‾Sn(xy)F‾Sn(xy)F‾Sn(x)F‾Sn(x)F‾S(n)(x)⩽max1⩽n⩽Klim supy↑1lim supx→∞F‾Sn(xy)F‾Sn(x).

According to Lemma 2, the d.f. FSn belongs to the class C for each fixed n. Therefore, lim supy↑1lim supx→∞I1⩽1.

Let us now consider the term I2 from expression (10). If x>0 0$]]>, then the numerator of I2 can be estimated as follows: ∑n=K+1∞P(S(n)>xy)P(η=n)= ∑n=K+1∞P(max{S1,…,Sn}>xy)P(η=n)⩽ ∑n=K+1∞P(max{S1(+),…,Sn(+)}>xy)P(η=n)= ∑n=K+1∞P(Sn(+)>xy)P(η=n), \mathit{xy})\mathbb{P}(\eta =n)\\{} & \displaystyle \hspace{1em}={\sum \limits_{n=K+1}^{\infty }}\mathbb{P}\big(\max \{S_{1},\dots ,S_{n}\}>\mathit{xy}\big)\mathbb{P}(\eta =n)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{n=K+1}^{\infty }}\mathbb{P}\big(\max \big\{{S_{1}^{(+)}},\dots ,{S_{n}^{(+)}}\big\}>\mathit{xy}\big)\mathbb{P}(\eta =n)\\{} & \displaystyle \hspace{1em}={\sum \limits_{n=K+1}^{\infty }}\mathbb{P}\big({S_{n}^{(+)}}>\mathit{xy}\big)\mathbb{P}(\eta =n),\end{array}\]]]> where Sk(+):=ξ1++⋯+ξk+ and ξ+:=ξ1[0,∞)(ξ) .

We can apply Lemma 3 for the last sum because of condition (b) of the theorem and the fact that Fξ1∈C ⇒ Fξ1+∈D . Using this lemma, we get ∑n=K+1∞P(S(n)>xy)P(η=n)⩽c3F‾ξ1(xy) ∑n=K+1∞np+1P(η=n), \mathit{xy})\mathbb{P}(\eta =n)\leqslant c_{3}\overline{F}_{\xi _{1}}(\mathit{xy}){\sum \limits_{n=K+1}^{\infty }}{n}^{p+1}\mathbb{P}(\eta =n),\]]]> for some positive constant c3 and for all x>0 0$]]> and y∈(0,1) .

On the other hand, for the denominator of I2 , we have P(S(η)>x)= ∑n=1∞P(S(n)>x)P(η=n)⩾P(max{S1,…,Sa}>x)P(η=a)⩾P(Sa>x)P(η=a). x)& \displaystyle ={\sum \limits_{n=1}^{\infty }}\mathbb{P}(S_{(n)}>x)\mathbb{P}(\eta =n)\\{} & \displaystyle \geqslant \mathbb{P}\big(\max \{S_{1},\dots ,S_{a}\}>x\big)\mathbb{P}(\eta =a)\\{} & \displaystyle \geqslant \mathbb{P}(S_{a}>x)\mathbb{P}(\eta =a).\end{array}\]]]>

According to the conditions of the theorem, the d.f. Fξk belongs to the class  C for each fixed index k∈N . Hence, using Lemma 4, we get lim infx→∞P(Sa>x)F‾ξ1(x)⩾lim infx→∞P(Sa>x)∑i=1aF‾ξi(x)lim infx→∞∑i=1aF‾ξi(x)F‾ξ1(x)=lim infx→∞∑i=1aF‾ξi(x)F‾ξ1(x)⩾lim infx→∞F‾ξ1(x)F‾ξ1(x)=1. x)}{\overline{F}_{\xi _{1}}(x)}& \displaystyle \geqslant \underset{x\to \infty }{\liminf }\frac{\mathbb{P}(S_{a}>x)}{{\textstyle\sum _{i=1}^{a}}\overline{F}_{\xi _{i}}(x)}\underset{x\to \infty }{\liminf }\frac{{\textstyle\sum _{i=1}^{a}}\overline{F}_{\xi _{i}}(x)}{\overline{F}_{\xi _{1}}(x)}\\{} & \displaystyle =\underset{x\to \infty }{\liminf }\frac{{\textstyle\sum _{i=1}^{a}}\overline{F}_{\xi _{i}}(x)}{\overline{F}_{\xi _{1}}(x)}\geqslant \underset{x\to \infty }{\liminf }\frac{\overline{F}_{\xi _{1}}(x)}{\overline{F}_{\xi _{1}}(x)}=1.\end{array}\]]]>

The last two estimates imply that P(S(η)>x)⩾12F‾ξ1(x)P(η=a), x)\geqslant \frac{1}{2}\overline{F}_{\xi _{1}}(x)\mathbb{P}(\eta =a),\]]]> for sufficiently large x.

Therefore, by inequalities (12) and (13) we have lim supy↑1lim supx→∞I2⩽2c3P(η=a)(lim supy↑1lim supx→∞F‾ξ1(xy)F‾ξ1(x)) ∑n=K+1∞np+1P(η=n).

Finally, substituting estimates (11) and (14) into (10), we obtain that lim supy↑1lim supx→∞P(S(η)>xy)P(S(η)>x)⩽1+2c3P(η=a)E(ηp+11[K+1,∞)(η)), \mathit{xy})}{\mathbb{P}(S_{(\eta )}>x)}\leqslant 1+\frac{2c_{3}}{\mathbb{P}(\eta =a)}\mathbb{E}\big({\eta }^{p+1}\mathbb{1}_{[K+1,\infty )}(\eta )\big),\]]]> for an arbitrary K⩾a .

This inequality implies the desired relation (9) due to condition (c) of the theorem. Theorem 5 is proved.  □

 Acknowledgments

We would like to thank the anonymous referees for the detailed and helpful comments on the first version of the manuscript and especially for the proposed new way of proving Theorem 4.

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