Fengyang CHENG Na LI

1 Introduction

Let{X,Xk:k≥1}be a sequence of random variables with a common distributionFand letτbe a nonnegative integer-valued random variable with a distributionFτ.For any distributionGand real numberx,we letand denote its tail byThe aim of the present paper is to investigate the asymptotic behavior of the tail probability of a random sumwhen the random numberτhas a heavier tail than the summands,Random sums play important roles in many applied probability fields such as financial insurance,risk theory,teletraffic,queueing theory and so on.Generally speaking,it is hard to obtain the precise distribution ofSτ,so one possible approachis to discuss the asymptotic behavior of the tail probabilityP(Sτ>x)asx→∞.

Hereafter,all limit relationships are forx→∞unless otherwise stated.For two positive functionsa(x)andb(x),we writea(x)∼b(x)if limand writea(x)=o(b(x))if lim

Next,we introduce some common distribution classes.A random variableXor its distributionFis said to be heavy-tailed if

for any positive numbert,and is otherwise light-tailed.Below we list some of the commonly used subclasses of heavy-tailed distributions.

A random variableXor its distributionFis said to be long-tailed(denoted byX∈orF∈)if=1 for any fixedy>0;to have a consistently varying tail(denoted byX∈CorF∈C)if=1;to have a dominatedly varying tail(denoted byX∈DorF∈D)if<∞for any fixedy∈(0,1);to have a regularly varying tail with an indexαfor someα>0(denoted byX∈R−αorF∈R−α),if=y−αfor any fixedy>0;and to be subexponential(denoted byifF∈Land=2,whereF∗2=F∗Fdenotes the convolution ofFwith itself.

It is well-known that

for anyα≥0.

In many areas of applied probability,it is found that random sums often have a heavy tail.Many researchers are interested in the questions of what causes the heavy tail of a random sum and what is the relationship among the tail probabilities ofSτ,Xandτ.In one case where the summands have a heavier tail thanit is found that the tail behavior of a random sumSτis decided by the tail ofXand the mean ofτ,and thatSτandXbelong to the same subclass of heavy-tailed distributions(see[4–5,8,10,13–14]etc.).

Recently,other cases in which the tail ofXis not heavier than that ofτhave attracted a lot of academic attention.Faÿ et al.[7]gave sufficient conditions forwhenXhas a lighter tail thanand gave necessary conditions forwhenXhas a lighter tail thanSτ.It states that the tail behavior of a random sumSτis decided by the tail ofτand the mean ofX,and thatSτandτbelong to the same classifSome sufficient conditions forSτ∈Chave been obtained by many authors(see[1,11,15]etc.).

The purpose of this paper is to give necessary and sufficient conditions forSτ ∈Cwhenin which the summandsare extended negatively dependent random variables(see Definition 2.1 below)defined on(−∞,∞).

We will introduce some definitions of the dependence structure and give the main results of this paper in Section 2.The proofs of the theorems are given in Section 3.

2 Main Results

First,we give some definitions of the dependence structure,which are introduced by Chen et al.[3]and Liu[9].

Definition 2.1(see[3,9])A finite family of random variables{Xk:1≤k≤n}is said to be

(1)lower extended negatively dependent(LEND for short)if there exists a constant M≥1,such that for all real numbers x1,x2,···,xn,

(2)upper extended negatively dependent(UEND for short)if there exists a constant M≥1,such that for all real numbers x1,x2,···,xn,

(3)extended negatively dependent(END for short)if there exists a constant M≥1,such that both(2.1)and(2.2)hold for all real numbers x1,x2,···,xn,···.

The constantMin equations(2.1)–(2.2)is said to be dominating constant.A sequence of random variables{Xk:k≥1}is said to be END(LEND,UEND)if each of its finite subfamilies is END(LEND,UEND)for some common dominating constantM.

The END structure covers many negative dependence structures and,more interestingly,it covers certain positive dependence structures.More detailed discussions and some examples can be found in Chen et al.[3]and Liu[9].

Now,we give the main results of this paper as follows.

Theorem 2.1Let{X,Xk:k≥1}be a sequence of END random variables with a common distribution F satisfying EX>0.Let τ be a nonnegative integer-valued random variable with a distribution Fτ,independent of{X,Xk:k≥1}.Suppose that one of the following two conditions holds:

(i)Eτ<∞and

or

(ii)Eτ=∞and

for some r≥1and δ>1and

Then the following two assertions are equivalent:

(a)∈C;

(b)Sτ∈C.

Furthermore,each of them implies that

Remark 2.1The following question naturally occurs:Can(2.6)imply(a)or(b)?

The following example gives a negative answer.

Example 2.1LetXbe degenerate atp>0(soF∉C)and letbe any nonnegative integer-valued random variable.Obviously,we have

This shows that(2.6)may not imply(a)or(b).

Remark 2.2IfFτ ∈R−αfor someα∈(0,1),then(2.5)holds for anyr≥1 by Karamata’s theorem(see[2,Propositions 1.5.8 and 1.5.9a])andEτ=∞.Remark 4.5 of Faÿ et al.[7]gave an example in whichandcan not imply(2.6)ifHence,some extra conditions are needed ifEτ=∞.It is obvious that both(1.4)in[15]and(3.11)in[11]are stronger than(2.5)whenEτ=∞.

Remark 2.3IfEτ<∞,then(2.3)implies thatEXI(X>0)<∞,whereI(·)is the indicator function of a set.IfEτ=∞,then(2.5)implies thatEXI(X>0)<∞.Hence,the conditions of Theorem 2.1 always imply thatE|X|<∞sinceEX>0.

3 Proof of Theorem 2.1

Before giving the proof of the main results,we first give several lemmas.The first lemma is a direct consequence of Definition 2.1 and was mentioned by Chen et al.[3].

Lemma 3.1If{Xk:1≤k≤n}are UEND(or LEND)random variables for some dominating constant M and{hk(·):1≤k≤n}are non-decreasing functions,then{hk(Xk):1≤k≤n}are still UEND(or LEND)random variables for the same dominating constant M.

Lemma 3.2Let{Xk:1≤k≤n}be UEND(or LEND)random variables for some dominating constant M.Let{Yk:1≤k≤n}be independent random variables,independent of{Xk:1≤k≤n}.Let

Then,{Zk:1≤k≤n}are UEND(or LEND)random variables for the same dominating constant M.

ProofWe only prove the case that{Xk:1≤k≤n}are UEND.For any real numbersx1,x2,···,xn,

The next lemma is a slight adjustment of Corollary 3.1 of Tang[12].

Lemma 3.3Let{X,Xk:k≥1}be a sequence of UEND random variables with a common distribution F∈D and a meanµ=EX.Then for each fixed γ>0and some C=C(γ)irrespective to x and n,the inequality

holds uniformly for all x≥γn and n=1,2,···.

ProofThe proofis just similar to that of Corollary 3.1 of Tang[12]and hence is omitted.

The following three lemmas play key roles in the proof of Theorem 2.1.

Lemma 3.4Let{X,Xk:k≥1}be a sequence of END random variables with a common distribution F satisfying EX∈(0,∞).Let τ be a nonnegative integer-valued random variable with a distribution Fτ,independent of{X,Xk:k≥1}.Then

for any c>1.

ProofFor anyc>1 andx>0,we have

For any fixedε>0,by Theorem 1 in[3],there existsx1>0,such that

holds for allx>x1andConsequently,for anyx>x1,it follows that

By the arbitrariness ofε,(3.1)holds for anyc>1.

Lemma 3.5Under the conditions of Theorem2.1,if Sτ∈L∩D,then we have

for any v<1.

ProofObviously,(3.2)holds ifv≤0,so we supposev∈(0,1)in the later discussion.By Remark 2.3,it follows thatE|X|<∞.It is easy to see that

If we can prove that

then from(3.3)and(3.4),it immediately follows that

whichis equivalent to(3.2).So we only need to prove(3.4).

First we discuss case(i)whereEτ<∞:By Lemma 3.4 andv∈(0,1),it follows that

Combining with(2.3)andSτ∈Dyields that

Let{Y,Yk:k≥1}be a sequence of independent identically distributed random variables with a common distributionV,whereVis the uniform distribution on the interval[0,1],independent of{X,Xk:k≥1}andτ.LetZ=X+Y,Zk=Xk+Yk,k≥1.Then,by Lemma 2.2,{Z,Zk:k≥1}is a sequence of END random variables with a common distributionF∗V.BySτ∈L∩Dand(2.3),it is easy to see thatP(Z>x)=o(P(Sτ>x)).By Lemma 4.4 in[7],there exists a nondecreasing slowly varying functionL(x)satisfying

Hence there existsx′>0,such that

holds for allx≥x′.Define a distributionGas follows:

and let

where

It is easy to see thatP(X′≤x)=G(x)for all real numberxand

By Proposition A.16(d)in[6],it follows thatfor allk≥1,which implies thata.s.for allk≥1 sinceG(x)≤F∗V(x)for all real numbersx.Moreover,it follows thatEX≤EX′<∞by the definition ofX′andEτ<∞.Write

Thenholds for allx≥0 andn≥1 sincea.s.holds for alln≥1.

For allx>0,we splitp(x)into two parts as

Note thatSτ∈∩DimpliesG∈∩D.By Lemma 3.3,there exists a positive constantC=C(v)independent ofxandn,such that

holds for allCombining with(3.5)we have

On the other hand,by Theorem 1 in[3],we have

It follows that

Hence(3.4)follows for the caseEτ<∞.

Now we discuss the case(ii)whereEτ=∞:Let

ifr>1;and let

ifr=1.The assumption(2.4)implies that

By Lemma 4.4 in[7],there exists a nondecreasing slowly varying functionL(x)satisfying

Thus there existsx′>1,such that

holds for allx≥x′.Define a distributionGas follows:

It is obvious thatG∈R−r⊂∩D.Without loss of generality,we assume thatFis absolutely continuous,otherwiseFcan be replaced byF∗V,whereVis the uniform distribution on the interval[0,1],so thenF∗Vis absolutely continuous andLet

and

Similarly to the proof of(3.7),there exists a positive constantC=C(v)independent ofxandn,such that

Hence,for sufficiently largex,we have

Combining with(3.6)and(3.8),(3.4)is obtained.

Lemma 3.6Under the conditions of Theorem2.1,if τ∈L∩D,then(3.2)holds for any v<1.

ProofThe proofis similar to Lemma 3.5 and hence is omitted.

Proof of Theorem 2.1Obviously,(a)and(2.6)imply(b);and(b)and(2.6)imply(a).Therefore,we need only to prove that either(a)or(b)implies(2.6).We first prove that(b)implies(2.6).It suffices to prove that

and

By Lemma 3.4,(3.1)holds for anyc>1.It follows that

sinceSτ∈C.(3.9)is obtained.

The proof of(3.10)is similar to that of(3.9).By Lemma 3.5,(3.2)holds for allv<1.It follows that

The proof of the fact that(a)implies(2.6)is quite similar to the above.By Lemma 3.4,Lemma 3.6 andFτ∈C,it follows that

and

This finishes the proof of Theorem 2.1.

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