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 tailexponential tailO-exponential tailrandom sumrandom convolutioninhomogeneous distributionsclosure property62E2060E0560F1044A35Introduction
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 SηFSη(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 ofO-exponential distributionsOLif0<lim infx→∞F‾(x+a)F‾(x)⩽lim supx→∞F‾(x+a)F‾(x)<∞for alla∈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)>00$]]> for all x∈R.
It is obvious that a d.f. F belongs to the class OL if and only if
lim supx→∞F‾(x−1)F‾(x)<∞
or, equivalently, if and only if
supx⩾0F‾(x−1)F‾(x)<∞.
The last condition shows that class OL is quite wide. We further describe some more popular subclasses of OL for which we will present some results on the random convolution of distributions from these subclasses.
A d.f. F is said to belong to the classLof long-tailed d.f.s if for every fixeda>00$]]>, we havelimx→∞F‾(x+a)F‾(x)=1.
A d.f. F is said to belong to the classL(γ)of exponential distributions with someγ>00$]]>if for any fixeda>00$]]>, we havelimx→∞F‾(x+a)F‾(x)=e−aγ.
A d.f. F belongs to the classD (or has a dominatingly varying tail) if for every fixeda∈(0,1), we havelim supx→∞F‾(xa)F‾(x)<∞.
A d.f. F supported on the interval[0,∞)belongs to the classS (or is subexponential) iflimx→∞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 classS∗ ( or is strongly subexponential) ifμ:=∫[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,…}. IfE(1+δ)η<∞for someδ>00$]]>, thenFSη∈S.
In the case of strongly subexponential d.f.s, the following result, which involves weaker restrictions on the r.v. η, can be derived from Theorem 1 of Denisov et al. [6] and Corollary 2.36 of Foss et al. [10].
Let{ξ1,ξ2,…}be independent copies of a nonnegative r.v. ξ with strongly subexponential d.f.Fξand finite meanEξ. Let η be a counting r.v. independent of{ξ1,ξ2,…}. IfP(η>x/c)=x→∞o(F‾ξ(x))x/c)\underset{x\to \infty }{=}o(\overline{F}_{\xi }(x))$]]>for somec>Eξ\mathbb{E}\xi $]]>, thenFSη∈S∗.
Similar results for classes D, L, and OL can be found in the papers of Leipus and Šiaulys [12] and Danilenko and Šiaulys [4]. We further present Theorem 6 from [12].
Let{ξ1,ξ2,…}be independent r.v.s with common d.f.Fξ∈L, and let η be a counting r.v. independent of{ξ1,ξ2,…}having d.f.Fη. IfF‾η(δx)=x→∞o(xF‾ξ(x))for eachδ∈(0,1), thenFSη∈L.
In all presented results, r.v.s {ξ1,ξ2,…} are identically distributed. In this work, we consider independent, but not necessarily identically distributed, r.v.s. As was noted, we restrict our consideration on the class OL. In fact, in this paper, we generalize the results of [4]. If {ξ1,ξ2,…} may be not identically distributed, then various collections of conditions on r.v.s {ξ1,ξ2,…} and η imply that FSη∈OL. The rest of the paper is organized as follows. In Section 2, we formulate our main results. In Section 3, we present all auxiliary assertions, and the detailed proofs of the main results are presented in Section 4. Finally, a few examples of O-exponential random sums are described in Section 5.
Main results
In this section, we formulate our main results. The first result describes the situation where the tails of d.f.s Fξk for large indices k are uniformly comparable with itself at the points x and x−1 for all x∈[0,∞).
Let{ξ1,ξ2,…}be independent nonnegative random variables with d.f.s{Fξ1,Fξ2,…}, and let η be a counting r.v. independent of{ξ1,ξ2,…}. ThenFSη∈OLif the following three conditions are satisfied.
For someκ∈supp(η)∖{0}={n∈N:P(η=n)>0}0\}$]]>,Fξκ∈OL.
For eachk∈supp(η),k⩽κ, eitherlimx→∞F‾ξk(x)F‾ξκ(x)=0orFξk∈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 sumFSηisO-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}. ThenFSη∈OLunder the following three conditions.
P(η⩽D)=1.
For someκ∈supp(η)∖{0},Fξκ∈OL.
For eachk∈{1,2,…,D}, eitherlimx→∞F‾ξk(x)F‾ξκ(x)=0orFξk∈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.Fηindependent of{ξ1,ξ2,…}. ThenFSη∈OLif the following five conditions are satisfied.
For some κ∈supp(η)∖{0}, Fξκ∈OL.
For eachk∈supp(η),k⩽κ, eitherlimx→∞F‾ξk(x)F‾ξκ(x)=0orFξk∈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 satisfyingF‾(x)>00$]]>,G‾(x)>00$]]>,x∈R. ThenF∗G‾(x−t)F∗G‾(x)⩽max{supy⩾vF‾(y−t)F‾(y),supy⩾x−v+tG‾(y−t)G‾(y)}for allx∈R,v∈R, andt>00$]]>.
The following assertion is the well-known Kolmogorov–Rogozin inequality for concentration functions. Recall that the Lévy concentration function or simply concentration function of a r.v. X is the function
QX(λ)=supx∈RP(x⩽X⩽x+λ),λ∈[0,∞).
The proof of the next lemma can be found in [14] (Theorem 2.15).
LetX1,X2,…,Xnbe independent r.v.s, and letZn=∑k=1nXk. Then, for alln∈N,QZn(λ)⩽Aλ{∑k=1nλk2(1−QXk(λk))}−1/2,where A is an absolute constant, and0<λk⩽λfor eachk∈{1,2,…,n}.
The following assertion describes sufficient conditions under which the d.f. of two independent r.v.s belongs to the class OL.
LetX1andX2be independent r.v.s with d.f.sFX1andFX2, respectively. Then the d.f.FX1∗FX2of the sumX1+X2isO-exponential ifFX1∈OLand one of the following two conditions holds:∙limx→∞F‾X2(x)F‾X1(x)=0,∙FX2∈OL.
We split the proof into three parts.
I. First, suppose that P(X2⩽D)=1 for some D>00$]]>. 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)>00$]]> 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}. LetFX1∈OLand suppose that, for eachk∈{2,3,…,n}, eitherlimx→∞F‾Xk(x)F‾X1(x)=0orFXk∈OL. Then the d.f.FX1∗FX2∗⋯∗FXnbelongs to the classOL.
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 Theorem4. 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)⩽c3x-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)=c3x)}{\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 Theorem5. 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>00$]]> 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 Theorem6. 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)⩽c4x-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>22$]]>, c6>00$]]> 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)⩽c7x-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 O-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>00$]]>. In addition, we suppose that the r.v. ξD+k for each k∈N is distributed according to the exponential law with parameter λ/k, that is,
F‾ξD+k(x)=e−λx/k,x⩾0.
It follows from Theorem 4 that the d.f. of the random sum Sη is O-exponential for each counting r.v. η independent of {ξ1,ξ2,…} under the condition P(η=κ)>00$]]> for some κ∈{1,2,…,D} because:
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:
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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