In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum Sν=X1+⋯+Xν of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of {X1,X2,…}. The conditions are provided for the relation P(Sν>x)∼EνP(X1>x) to hold, as x→∞, involving the finiteness of E|X1|. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment E|X1|r for some r>1 was assumed.

Heavy-tailed distributionConsistently varying distributionRandomly stopped sum60E0560F1060G40Introduction and preliminaries

Let X,X1,X2,… be independent identically distributed (i.i.d.) random variables (r.v.s). Denote a sequence of partial sums {Sn,n⩾0} by
S0=0,Sn:=X1+⋯+Xn,n⩾1.
In this paper, we consider the randomly stopped sumSν=X1+⋯+Xν,
where ν is a counting r.v. taking values in N0:={0,1,2,…}. We assume that ν is nondegenerate at zero, i.e. P(ν=0)<1, and that ν is independent of {X,X1,X2,…}. Denote FX, Fν and FSν the distributions of X, ν and Sν, respectively. In the case where primary r.v.s are heavy-tailed and nonnegative, the standard result states that if ν has finite mean Eν, and its distribution tail is lighter than the tail of X, then
FSν‾(x)∼x→∞EνFX‾(x).
For important contributions, see Stam [15], Daley et al. [4], Embrechts et al. [7], Faÿ et al. [8]. In Section 4 of the last paper, the case of nonnegative regularly varying summands was examined in detail. Note that, generally, relationship (2) can be obtained under different conditions on r.v.s X and ν (see, e.g., Daley et al. [4]). More precisely, (2) holds under various heavy-tailed distribution classes, moment conditions on X and ν, relationships between the distribution tails FX‾ and Fν‾ (typically, Fν‾(x)=o(FX‾(x))). Usually, weakening the conditions on Fν, stronger conditions on FX are assumed. In the case of real-valued r.v.s, the conditions for (2) depend also on the sign of the mean μ=EX if it exists. For instance, in the case of negative mean, the relation (2) holds for all subclasses of S∗ (see definition below), which includes most subexponential distributions with finite mean, see Denisov et al. [6, Theorem 1].

In this paper, we pose the problem under what ‘minimal’ moment conditions relation (2) holds in the case of real-valued consistently varying distribution FX with zero mean. Recall that the class of consistently varying distributions (see definition below) contains the regularly varying class of distributions.

Before formulating and discussing the main result of the paper, we will introduce the related subclasses of heavy-tailed distributions, some notions, and known results. We will say that a distribution F=1−F‾ is on R:=(−∞,∞) if F‾(x)>0 for all x∈R. All limiting relations are assumed as x→∞ unless it is stated to the contrary. For two eventually positive functions a(x) and b(x), a(x)∼b(x) means that lima(x)/b(x)=1; a(x)≍b(x) means that 0<lim infa(x)/b(x)⩽lim supa(x)/b(x)<∞. We denote a+:=max{a,0}, a−:=−min{a,0}.

A distribution F on R is said to be heavy-tailed, denotedF∈H, if its Laplace–Stieltjes transform satisfies∫−∞∞eδxdF(x)=∞for anyδ>0.Otherwise, F is said to be light-tailed.

Next we introduce the heavy-tailed distribution subclasses which will be used in the paper.

A distribution F on R is said to be regularly varying with indexα⩾0and denotedF∈R(α)if its tail satisfieslimF‾(xy)F‾(x)=y−αfor anyy>0.F∈R(0)is said to be slowly varying distribution.

A distribution F on R is said to be consistently varying, denoted byF∈C, iflimy↘1lim infx→∞F‾(xy)F‾(x)=1.

A distribution F on R is said to belong to dominatedly varying class of distributions, denotedF∈D, iflim supF‾(xy)F‾(x)<∞for all (or, equivalently, for some)y∈(0,1).

It holds that C⊂D⊂H.

Next, for a distribution F on R denote
F‾∗(y):=lim infx→∞F‾(xy)F‾(x),F‾∗(y):=lim supx→∞F‾(xy)F‾(x),y>1,
and introduce the upper and lower Matuszewska indices by equalities
JF+=−limy→∞logF‾∗(y)logy,JF−=−limy→∞logF‾∗(y)logy.
Clearly, 0⩽JF−⩽JF+⩽∞. It is well known that F∈D if and only if JF+<∞.

SetF+(x):=F(x)1{x⩾0}. A distribution F on R is said to be subexponential and denotedF∈SifF+∗F+‾(x)∼2F‾(x).

Note that F∈S implies
F∗n‾(x)∼nF‾(x)for alln⩾2,
see, e.g., Foss et al. [9, Corollary 3.20].

A distribution F on R withmF:=∫0∞F‾(u)du∈(0,∞)belongs toS∗(or is strong subexponential) if∫0xF‾(x−y)F‾(y)dy∼2mFF‾(x).

It holds that C⊂S∗⊂S provided the mean is finite.

More details on the mentioned heavy-tailed classes can be found in the recent book [11].

First, we formulate some known results for class C.

LetX,X1,X2,…be i.i.d. real-valued r.v.s with the common distributionFX∈Cand let ν be an independent counting r.v. Let either of conditions hold:

Eνp+1<∞for somep>JFX+, or

E|X|<∞,Fν‾(x)=o(FX‾(x)), or

E|X|<∞,Eν<∞andEX<0.

ThenFSν‾(x)∼EνFX‾(x).

Proof. In the case of nonnegative r.v.s, part (a) of the proposition can be found in Leipus and Šiaulys [10, Corollary 3]. We provide a short proof for the case of distributions on R. Write
FSν‾(x)=∑n=1∞FX∗n‾(x)P(ν=n),x⩾0.
Since C⊂S, we have FX∗n‾(x)∼nFX‾(x) for any n⩾1. In addition, as C⊂D, according to Theorem 3 in Daley et al. [4], for any p>JFX+, there exists a finite positive constant C, independent of x and n, such that
supx∈RFX∗n‾(x)FX‾(x)⩽Cnp+1.
This implies (5) by the dominated convergence theorem. Part (b) can be found in Ng et al. [13, Theorem 2.3] or Denisov et al. [6, Corollary 3]. Part (c) follows from Denisov et al. [6, Theorem 1] and relationship C⊂S∗ (in the case of regularly varying distributions, see Borovkov and Borovkov [2, Theorem 7.1.1]). □

We will focus our attention to the case where EX=0 and show that in this case the result in part (b) can be improved replacing o(·)-condition to O(·)-condition. Note that, in the case of zero mean and in the more general setup, Olvera-Cravioto [14, Theorem 2.1 (b)] obtained the following result.

LetX,X1,X2,…be i.i.d. real-valued r.v.s with the common distributionFX∈C, and let ν be an independent counting r.v. Assume thatJFX−>0,E|X|r<∞for somer>1,EX=0andFν‾(x)=O(FX‾(x)). Then (5) holds.

As noted by Olvera-Cravioto [14], the proof of the result follows the standard heavy-tailed techniques from Nagaev [12], Borovkov [1] (see also Borovkov and Borovkov [2]), based on the exponential bounds for sums of truncated r.v.s. Moreover, it was conjectured that the requirement E|X|r<∞, r>1 might be weakened with a different proof technique.

In our paper we prove that the result of Proposition 2 indeed holds under the condition E|X|<∞, accordingly modifying the proof. Specifically, some ideas from Cline and Hsing [3], Tang [16] and Danilenko and Šiaulys [5] have been used in the proof of the main result. Apparently, the alternative proof of the main result can be constructed using the bounds in Theorem 1 of Tang and Yan [18].

Main results

LetX,X1,X2,…be i.i.d. r.v.s with the distributionFX∈C, and let ν be an independent counting r.v. IfE|X|<∞,EX=0,JFX−>0, andFν‾(x)=O(FX‾(x)), then (5) holds.

Observe that conditions E|X|<∞ and Fν‾(x)=O(FX‾(x)) imply finiteness of the moment Eν<∞. The statement of the theorem follows from Propositions 3 and 4 below in which the upper and lower asymptotic bounds are obtained.

Note that, in the case of dominatedly varying distribution FX with finite mean, the condition Fν‾(x)=O(FX‾(x)) (both for μ>0 and μ⩽0) is sufficient for the relationship FSν‾(x)≍FX‾(x) (see, e.g., Leipus and Šiaulys [10], Yang and Gao [19]). Taking into account the closure of class D under weak tail equivalence, this yields that the distribution of random sum Sν is in D.

LetX,X1,X2,…be i.i.d. r.v.s with the common distributionFX∈S, and let ν be an independent counting r.v. with finite meanEν. Thenlim infFSν‾(x)EνFX‾(x)⩾1.

Under the conditions of Theorem1,lim supFSν‾(x)EνFX‾(x)⩽1.

From the main theorem we obtain the following statement for regularly varying distributions. To the best of our knowledge, this is a new result.

LetX,X1,X2,…be i.i.d. r.v.s with the distributionFX∈R(α),α⩾1, and let ν be an independent counting r.v. IfE|X|<∞,EX=0, andFν‾(x)=O(FX‾(x)), then (5) holds.

Remark that if FX∈R(α), α>1, then the condition E|X|<∞ is automatically satisfied.

Proof of Proposition <xref rid="j_vmsta236_stat_005">3</xref>

For K∈N and large x we have
FSν‾(x)⩾P(Sν>x,ν⩽K)=∑n=1KFX∗n‾(x)P(ν=n).
Since FX∗n‾(x)∼nFX‾(x), we get that
lim infFSν‾(x)EνFX‾(x)⩾1Eν∑n=1KnP(ν=n)=Eν1{ν⩽K}Eν.
The assertion of the proposition follows now from the last estimate by passing K to infinity. □

Proof of Proposition <xref rid="j_vmsta236_stat_006">4</xref>

Let K∈N and δ∈(0,1) be temporarily fixed numbers. For sufficiently large x we have
FSν‾(x)=P(Sν>x,ν⩽K)+P(Sν>x,K<ν⩽xδ−1)+P(Sν>x,ν>xδ−1)=∑n=1KP(Sn>x)P(ν=n)+∑K<n⩽xδ−1P(Sn>x,∪k=1n{Xk>x(1−δ)})P(ν=n)+∑K<n⩽xδ−1P(Sn>x,∩k=1n{Xk≤x(1−δ)})P(ν=n)+P(Sν>x,ν>xδ−1)⩽∑n=1KFX∗n‾(x)P(ν=n)+∑K<n⩽xδ−1nFX‾(x(1−δ))P(ν=n)+∑K<n⩽xδ−1P(∑k=1nXˆk>x)P(ν=n)+Fν‾(xδ−1)=:J1+J2+J3+J4,
where Xˆk:=min{Xk,x(1−δ)}.

Since FX∈C⊂S, it holds that FX∗n‾(x)∼nFX‾(x) for any fixed n. Therefore,
J1⩽(1+δ)FX‾(x)Eν1{ν⩽K}
for sufficiently large x⩾x1(K,δ).

In addition, J2⩽FX‾(x(1−δ))Eν1{ν>K},J3⩽Eνmaxn⩽xδ−1P(∑k=1nXˆk>x)n. Using the bound in Lemma 1 (i) for the class D and the condition Fν‾(x)=O(FX‾(x)), we obtain
J4=Fν‾(xδ−1)FX‾(xδ−1)FX‾(xδ−1)FX‾(x)FX‾(x)⩽c1FX‾(xδ−1)FX‾(x)FX‾(x)⩽c2δJFX−/2FX‾(x)
for large x⩾x2(δ) with some positive constants c1 and c2.

Substituting estimates (8)–(11) into (7), we obtain
FSν‾(x)EνFX‾(x)⩽max{J1FX‾(x)Eν1{ν⩽K},J2FX‾(x)Eν1{ν>K}}+J3FX‾(x)Eν+J4FX‾(x)Eν⩽max{1+δ,FX‾(x(1−δ))FX‾(x)}+maxn⩽xδ−1P(∑k=1nXˆk>x)nFX‾(x)+c2EνδJFX−/2
for x⩾max{x1(K,δ),x2(δ)}. Therefore,
lim supFSν‾(x)EνFX‾(x)⩽max{1+δ,lim supFX‾(x(1−δ))FX‾(x)}+lim supFX‾(x(1−δ))FX‾(x)lim supmaxn⩽xδ−1P(∑k=1nXˆk>x)nFX‾(x(1−δ))+c2EνδJFX−/2=max{1+δ,lim supFX‾(x(1−δ))FX‾(x)}+c2EνδJFX−/2
according to Lemma 2. The desired upper bound is then obtained taking δ↘0.

Auxiliary lemmas

The first auxiliary lemma can be found in Tang and Tsitsiashvili [17, Lemma 3.5].

Let the distributionF∈Dwith lower and upper Matuszewska indicesJF−andJF+, respectively.

IfJF−>0, then for any0⩽p1<JF−there exist positive constantsC1=C1(p1)andD1=D1(p1), such thatF‾(y)F‾(x)⩾C1(xy)p1for allx⩾y⩾D1.

For anyp2>JF+⩾0there exist positive constantsC2=C2(p2)andD2=D2(p2), such thatF‾(y)F‾(x)⩽C2(xy)p2for allx⩾y⩾D2.

For anyp>JF+it holds thatx−p=o(F‾(x)).

The following lemma is crucial in the proof of Proposition 4.

LetX,X1,X2,…be i.i.d. real-valued r.v.s with the dominatedly varying distributionFX∈D. IfE|X|<∞,EX=0, then, for anyδ∈(0,1),limmaxn⩽xδ−1P(∑k=1nXˆk>x)nFX‾(x(1−δ))=0,whereXˆk:=min{Xk,x(1−δ)}.

Proof. For any δ∈(0,1), set
a=a(x,n):=max{log1nFX‾(x(1−δ)),1},x∈R,n∈N.
The assumption E|X|<∞ implies that xFX‾(x(1−δ))→0 as x→∞. Since a(x,n) is nonincreasing in n, we get that for any δ∈(0,1)minn⩽xδ−1a(x,n)⩾log1xδ−1FX‾(x(1−δ))→∞
and a(x,n)=log(1/(nFX‾(x(1−δ)))) for large x (x⩾x3(δ)) and n⩽xδ−1.

By the exponential Markov inequality, for any h,x>0, we have
P(∑k=1nXˆk>x)⩽e−hxEexp{h∑k=1nXˆk}=e−hx(1+E(ehXˆ1−1))n.
Thus, by inequality 1+z⩽ez, z∈R,
P(∑k=1nXˆk>x)nFX‾(x(1−δ))⩽exp{−hx+a+nE(ehXˆ1−1)}.
Split the expectation E(ehXˆ1−1) as follows:
E(ehXˆ1−1)=K1+K2+K3+K4,
where
K1:=∫(−∞,0](ehu−1)dFX(u),K2:=∫(0,x(1−δ)a−2](ehu−1)dFX(u),K3:=∫(x(1−δ)a−2,x(1−δ)](ehu−1)dFX(u),K4:=(ehx(1−δ)−1)FX‾(x(1−δ)).
The inequalities |ez−1|⩽|z|, |ez−z−1|⩽z2/2, z⩽0, imply that
K1=hEX1{X⩽0}+E(ehX−hX−1)1{X⩽0}=−hEX−+E(ehX−1)1{X⩽−h−1/4}−hEX1{X⩽−h−1/4}+E(ehX−hX−1)1{−h−1/4<X⩽0}⩽−hEX−+2hE|X|1{X⩽−h−1/4}+h3/22.
The inequality ez−1⩽zez, z⩾0, implies that
K2⩽hehx(1−δ)a−2∫(0,x(1−δ)a−2]udFX(u)⩽hehx(1−δ)a−2EX+.
In addition, observe that
K3,K4⩽ehx(1−δ)FX‾(x(1−δ)a−2).
Substituting estimates (17), (18), (19) into (15)–(16), we get
P(∑k=1nXˆk>x)nFX‾(x(1−δ))⩽exp{2nehx(1−δ)FX‾(x(1−δ)a−2)}×exp{−hx+a+nh(2E|X|1{X⩽−h−1/4}+h1/22−EX−+ehx(1−δ)a2EX+)}.

According to Lemma 1 (iii), (x(1−δ))pFX‾(x(1−δ))→∞ for any p>JFX+. Hence, for large x (x⩾x4(δ,p)>x3(δ)),
max1⩽n⩽xδ−1a(x,n)⩽log(x(1−δ))pFX‾(x(1−δ))(x(1−δ))p⩽plog(x(1−δ)).
This relation implies that
min1⩽n⩽xδ−1x(1−δ)a−2→∞
and, since FX∈D, by Lemma 1 (ii), it holds
FX‾(x(1−δ)a−2)FX‾(x(1−δ))⩽c3a2p
for any p>JFX+, large x(x⩾x5(δ,p)>x4(δ,p)) and some positive constant c3=c3(δ,p).

Therefore, by condition EX=EX+−EX−=0, we get
P(∑k=1nXˆk>x)nFX‾(x(1−δ))⩽exp{2c3na2pehx(1−δ)FX‾(x(1−δ))}×exp{−hx+a+nh(2E|X|1{X⩽−h−1/4}+h1/22+(ehx(1−δ)a2−1)EX+)}=:P1P2
for h>0, n⩽xδ−1 and large x (x⩾x5(δ,p)).

Now, for x>0 set
h=h(x,n):=max{a(x,n)−2ploga(x,n)x(1−δ),1x(1−δ)}.
By (14), for large x (x⩾x6(δ,p)>x5(δ,p)),
h=a−2plogax(1−δ).
Hence, from (20) we obtain, that for x⩾x6(δ,p)maxn⩽xδ−1h(x,n)⩽maxn⩽xδ−1a(x,n)x(1−δ)⩽plog(x(1−δ))x(1−δ)→0.

With this choice of h, we obtain that, for large x(x⩾x6(δ,p)) and any n⩽xδ−1,
P1=exp{2c3na2pea−2plogaFX‾(x(1−δ))}=exp{2c3eanFX‾(x(1−δ))}=e2c3.
For P2, we have for large x and n⩽xδ−1P2=exp{−aδ1−δ+2ploga1−δ+na−2plogax(1−δ)(2E|X|1{X⩽−h−1/4}+h1/22+(e(a−2ploga)a−2−1)EX+)}⩽exp{−aδ1−δ+2ploga1−δ+aδ(1−δ)(2E|X|1{X⩽−h−1/4}+h1/22+(e1/a−1)EX+)}.
Since, by (14) and (23), minn⩽xδ−1a(x,n)→∞ and maxn⩽xδ−1h(x,n)→0, for large x (x⩾x7(δ,p)>x6(δ,p)), it holds that
maxn⩽xδ−1(2E|X|1{X⩽−h−1/4}+h1/22+(e1/a−1)EX+)⩽δ22.
Substituting this bound into (25), we obtain that, for large x,
maxn⩽xδ−1P2⩽maxn⩽xδ−1exp{−aδ1−δ+2ploga1−δ+aδ2(1−δ)}=maxn⩽xδ−1exp{−aδ−4ploga2(1−δ)}→0.
This, together with (22) and (24), implies the statement of the lemma. □

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