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 .

Let {ξ1,ξ2,…} be i.i.d. nonnegative r.v.s with common d.f. Fξ∈D and finite mean Eξ . Let η be a counting r.v. independent of {ξ1,ξ2,…} with d.f. Fη and finite mean Eη . Then d.f. FSη∈D iff   min{Fξ,Fη}∈D .

Let {ξ1,ξ2,…} be independent r.v.s, and η be a counting r.v. independent of {ξ1,ξ2,…} with d.f. Fη . Then the d.f.  FSη∈L if the following five conditions are satisfied: (i)

P(η⩾κ)>0 0$]]> for some κ∈N ;

Let r.v.s {ξ1,ξ2,…} be nonnegative independent, not necessarily 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∈D for some κ∈supp(η) ,

Let {ξ1,ξ2,…,ξD} , D∈N , be independent real-valued r.v.s, and η be a counting r.v. independent of {ξ1,ξ2,…,ξD} . Then the d.f. FSη belongs to the class C if the following conditions are satisfied: (a)

P(η⩽D)=1 ,

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 ,

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

Let {ξ1,ξ2,…} be independent nonnegative 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 ,

Let {ξ1,ξ2,…} be i.i.d. nonnegative r.v.s with common d.f. Fξ∈C , and η be a counting r.v. independent of {ξ1,ξ2,…} . Then the d.f. FSη belongs to the class C under the following two conditions: Eξ<∞ and F‾η(x)=o(F‾ξ(x)) .

Let {ξ1,ξ2,…} be independent r.v.s such that ξk are exponentially distributed for all even k, that is, F‾ξk(x)=e−x,x⩾0,k∈{2,4,6,…}, whereas, for each odd k, ξk is a copy of the r.v. (1+U)2G, where U and G are independent r.v.s, 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,…. In addition, let η be a counting r.v. independent of {ξ1,ξ2,…} and distributed according to the Poisson law.

Let {ξ1,ξ2,…} be independent r.v.s such that ξk are distributed according to the Pareto law (with tail index α=2 ) for all odd k, and ξk are exponentially distributed (with parameter equal to 1) for all even k, that is, F‾ξk(x)=1x2,x⩾1,k∈{1,3,5,…},F‾ξk(x)=e−x,x⩾0,k∈{2,4,6,…}. In addition, let η be a counting r.v independent of {ξ1,ξ2,…} that has the Zeta distribution with parameter 4, that is, P(η=m)=1ζ(4)1(m+1)4,m∈N0, where ζ denotes the Riemann zeta function.

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).\]]]>

Let {X1,X2,…Xn} be independent real-valued r.v.s. Assume that F‾Xi/F‾(x)→x→∞bi for some subexponential d.f. F and some constants bi⩾0 , i∈{1,2,…n} . Then FX1∗FX2∗⋯∗FXn‾(x)F‾(x)→x→∞ ∑i=1nbi.

Let {X1,X2,…,Xn},n∈N , be independent real-valued r.v.s. Then the d.f. FΣn of the sum Σn=X1+X2+⋯+Xn belongs to the class C if the following conditions are satisfied: (a)

FX1∈C ,

Evidently, we can suppose that n⩾2 . We split our proof into two parts.

First part. Suppose that FXk∈C for all k∈{1,2,…,n} . In such a case, the lemma follows from Lemma 1 and the inequality (3) a1+a2+⋯+amb1+b2+⋯+bm⩽max{a1b1,a2b2,…,ambm} for ai⩾0 and bi>0 0$]]>, i=1,2,…,m .

Namely, using the relation of Lemma 1 and estimate (3), we get that lim supx→∞F‾Σn(xy)F‾Σn(x)=lim supx→∞∑k=1nF‾Xk(xy)∑k=1nF‾Xk(x)⩽max1⩽k⩽nlim supx→∞F‾Xk(xy)F‾Xk(x) for arbitrary y∈(0,1) .

Since FXk∈C for each k, the last estimate implies that the d.f. FΣn has a consistently varying tail, as desired.

Second part. Now suppose that FXk∉C for some of indexes k∈{2,3,…,n} . By the conditions of the lemma we have that F‾Xk(x)=o(F‾X1(x)) for such k. Let K⊂{2,3,…,n} be the subset of indexes k such that FXk∉CandF‾Xk(x)=o(F‾X1(x)). By Lemma 2, F‾Σˆn(x)∼F‾X1(x), where Σˆn=X1+∑k∈KXk. Hence, (4) lim supx→∞F‾Σˆn(xy)F‾Σˆn(x)=lim supx→∞F‾X1(xy)F‾X1(x) for every y∈(0,1) .

Equality (4) implies immediately that the d.f. FΣˆn belongs to the class C . Therefore, the d.f. FΣn also belongs to the class C according to the first part of the proof because Σn=Σˆn+∑k∉KXk and FXk∈C for each k∉K . The lemma is proved.  □

Let {X1,X2,…} be independent real-valued r.v.s, and FXν∈D for some ν⩾1 . Suppose, in addition, 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 (5) F‾Sn(x)⩽c1np+1F‾Xν(x) for all n⩾ν and x⩾0 .

It suffices to prove that (6) lim supy↑1lim supx→∞F‾Sη(xy)F‾Sη(x)⩽1. According to estimate (3), for x>0 0$]]> and y∈(0,1) , we have F‾Sη(xy)F‾Sη(x)=∑n=1n∈supp(η)DP(Sn>xy)P(η=n)∑n=1n∈supp(η)DP(Sn>x)P(η=n)⩽max1⩽n⩽Dn∈supp(η)P(Sn>xy)P(Sn>x). xy)\mathbb{P}(\eta =n)}{{\textstyle\sum _{\begin{array}{c} n=1\\{} n\in \mathrm{supp}(\eta )\end{array}}^{D}}\mathbb{P}(S_{n}>x)\mathbb{P}(\eta =n)}\leqslant \underset{\begin{array}{c} 1\leqslant n\leqslant D\\{} n\in \mathrm{supp}(\eta )\end{array}}{\max }\frac{\mathbb{P}(S_{n}>xy)}{\mathbb{P}(S_{n}>x)}.\end{array}\]]]> Hence, by Lemma 3, lim supy↑1lim supx→∞F‾Sη(xy)F‾Sη(x)⩽lim supy↑1lim supx→∞max1⩽n⩽Dn∈supp(η)F‾Sn(xy)F‾Sn(x)⩽max1⩽n⩽Dn∈supp(η)lim supy↑1lim supx→∞F‾Sn(xy)F‾Sn(x)=1, which implies the desired estimate (6). The theorem is proved.  □

As in Theorem 5, it suffices to prove inequality (6). For all K∈N and x>0 0$]]>, we have P(Sη>x)=( ∑n=1K+ ∑n=K+1∞)P(Sn>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, for x>0 0$]]> and y∈(0,1) , we have (7) P(Sη>xy)P(Sη>x)=∑n=1KP(Sn>xy)P(η=n)P(Sη>x)+∑n=K+1∞P(Sn>xy)P(η=n)P(Sη>x)=:J1+J2. xy)}{\mathbb{P}(S_{\eta }>x)}& \displaystyle =\frac{{\textstyle\sum _{n=1}^{K}}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle \hspace{1em}+\hspace{2.5pt}\frac{{\textstyle\sum _{n=K+1}^{\infty }}\mathbb{P}(S_{n}>xy)\mathbb{P}(\eta =n)}{\mathbb{P}(S_{\eta }>x)}\\{} & \displaystyle =:\mathcal{J}_{1}+\mathcal{J}_{2}.\end{array}\]]]>

The random variable η is not degenerate at zero, so there exists a∈N such that P(η=a)>0 0$]]>. If K⩾a , then using inequality (3), we get J1⩽∑n=1n∈supp(η)KP(Sn>xy)P(η=n)∑n=1n∈supp(η)KP(Sn>x)P(η=n)⩽max1⩽n⩽Kn∈supp(η)P(Sn>xy)P(Sn>x). xy)\mathbb{P}(\eta =n)}{{\textstyle\sum _{\begin{array}{c}n=1\\{}n\in \mathrm{supp}(\eta )\end{array}}^{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}>xy)}{\mathbb{P}(S_{n}>x)}.\]]]> Similarly as in the proof of Theorem 5, it follows that (8) lim supy↑1lim supx→∞J1⩽max1⩽n⩽Kn∈supp(η)lim supy↑1lim supx→∞F‾Sn(xy)F‾Sn(x)=1. Since C⊂D , we can use Lemma 4 for the numerator of J2 to obtain ∑n=K+1∞P(Sn>xy)P(η=n)⩽c3F‾ξ1(xy) ∑n=K+1∞np+1P(η=n) xy)\mathbb{P}(\eta =n)\leqslant c_{3}\overline{F}_{\xi _{1}}(xy){\sum \limits_{n=K+1}^{\infty }}{n}^{p+1}\mathbb{P}(\eta =n)\]]]> with some positive constant c3 . For the denominator of J2 , we have that P(Sη>x)= ∑n=1∞P(Sn>x)P(η=n)⩾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}(S_{a}>x)\mathbb{P}(\eta =a).\end{array}\]]]>

The conditions of the theorem imply that Sa=ξ1+∑k∈Kaξk+∑k∉Kaξk, where Ka={k∈{2,…,a}:Fξk∉C,F‾ξk(x)=o(F‾ξ1(x))} .

By Lemma 2 F‾Sˆa(x)/F‾ξ1(x)→x→∞1, where FSˆa is the d.f. of the sum Sˆa=ξ1+∑k∈Kaξk In addition, by Lemma 3 we have that the d.f. FSˆa belongs to the class C .

If k∉Ka , then Fξk∈C by the conditions of the theorem. This fact and Lemma 1 imply that lim infx→∞P(Sa>x)F‾ξ1(x)⩾1+∑k∉Kalim infx→∞F‾ξk(x)F‾ξ1(x). x)}{\overline{F}_{\xi _{1}}(x)}\geqslant 1+\sum \limits_{k\notin \mathcal{K}_{a}}\underset{x\to \infty }{\liminf }\frac{\overline{F}_{\xi _{k}}(x)}{\overline{F}_{\xi _{1}}(x)}.\]]]> Hence, (9) P(Sη>x)⩾12F‾ξ1(x)P(η=a) x)\geqslant \frac{1}{2}\overline{F}_{\xi _{1}}(x)\mathbb{P}(\eta =a)\]]]> for x sufficiently large. Therefore, (10) lim supy↑1lim supx→∞J2⩽2c3P(η=a)(lim supy↑1lim supx→∞F‾ξ1(xy)F‾ξ1(x)) ∑n=K+1∞np+1P(η=n). Estimates (7), (8), and (10) imply that lim supy↑1lim supx→∞P(Sη>xy)P(Sη>x)⩽1+2c3P(η=a)Eηp+11{η>K} xy)}{\mathbb{P}(S_{\eta }>x)}\leqslant 1+\frac{2\hspace{0.1667em}c_{3}}{\mathbb{P}(\eta =a)}\mathbb{E}{\eta }^{p+1}\mathbb{1}_{\{\eta >K\}}\]]]> for arbitrary K⩾a .

Letting K tend to infinity, we get the desired estimate (6) due to condition (d). The theorem is proved.  □

Once again, it suffices to prove inequality (6).

By condition (e) we have that there exist two positive constants c4 and c5 such that ∑i=1nF‾ξi(x)⩽c5nF‾ξ1(x),x⩾c4,n∈N. Therefore, (11) ESn= ∑j=1nEξj= ∑j=1n(∫0c4+∫c4∞)F‾ξj(u)du⩽c4n+c5nEξ1=:c6n for a positive constant c6 and all n∈N .