Precise Large Deviations for Subexponential Distributions in a Multi Risk Model

Abstract

The precise large deviations asymptotics for the sums of independent identical random variables when the distribution of the summand belongs to the class $S^{\ast }$ of heavy tailed distributions is studied. Under mild conditions, we extend the previous results from the paper Denisov et al. (2010) to asymptotics that are valid uniformly over some time interval. Finally, we apply the main result on the multi-risk model introduced by Wang and Wang (2007).

1. Introduction

In this paper, the precise large deviations for a random walk whose steps represent random variables with distribution F from a subclass ${\mathcal{S}}^{\ast }$ of the subexponential class $\mathcal{S}$ is studied. What that means is F has heavy tail and is regular enough in order to exist the limit

$$\begin{array}{c}\underset{x\to \infty }{lim}{\displaystyle \frac{1}{\overline{F}(x)}}{\int }_0^x\overline{F}(x-y)\overline{F}(y)dy.\end{array}$$

Hence, the most popular distributions with heavy tails belong to the class ${\mathcal{S}}^{\ast }$, among others Pareto, Burr, Cauchy, Lognormal and Weibull. The inclusion of the class ${\mathcal{S}}^{\ast }$ in the class of subexponential distribution is proved proper, namely have been found subexponential distributions that do not belong to ${\mathcal{S}}^{\ast }$ (see Denisov et al. (2004)).

The topic of large deviations of non-random sums has already been well studied. Overviews are given in Nagaev (1973), Cline and Hsing (1991), Nagaev (1979). More insight in the field for non-random sums can be found in Heyde (1967a), Heyde (1967b), Heyde (1968), Nagaev (1969a), Nagaev (1969b), Wang et al. (2006). A general treatment of large deviations for subexponential distributions was presented in Pinelis (1985). Other contributions on precise large deviations are Konstantinides and Loukissas (2010), Loukissas (2012), Wang and Wang (2012). Papers Klüppelberg and Mikosch (1997), Tang et al. (2001) are studying large deviations of random sums. A review on large deviations for random sums is given in Mikosch and Nagaev (1998) and Mikosch and Nagaev (2001).

In case of independent r.v.’s, the large deviation has been established in Paulauskas and Skučaitė (2003) and Skučaitė (2004). Due to its importance in applications (see for example Issaka and SenGupta (2017), Habtemicael and SenGupta (2014)) this issue became very popular recently. Recent contributions in the topic are found in Baltrunas et al. (2004), Borovkov and Mogulskii (2006), Hult et al. (2005), Jelenkovic and Momcilovic (2004), Konstantinides and Mikosc (2005), Ng et al. (2004), Tang (2006), Chen et al. (2011), Yang et al. (2012), Gao et al. (2018), Zhang and Cheng (2017), Yang and Sha (2016).

Let us denote by $S_n$ the sum of n independent identically distributed random variables $X_1+\cdots +X_n$, with common distribution F. In the principle of one big jump, we get the intuition that indicates in the case of heavy tails, the most probable way that the event $\{S_n>x\}$ happens. Namely, only one of the random variables $X_1,\dots ,X_n$ becomes large while the others remain small. Asymptotically, as $x\to \infty$, we get $\mathbf{P}\left[S_n>x\right]\sim n\overline{F}(x)$, where by $\overline{F}$ is denoted the tail of the distribution F.

The multi-risk model was firstly introduced in Wang and Wang (2007) and has arisen from the following construction: Let $\{X_{i,j},j\ge 1\}$, $i=1,2,\dots ,k$ be i.i.d. non-negative random variables with common distribution function $F_i(x)$ and finite mean. Taking into account the notations

$$\begin{array}{c}{S}_{n_i}={\displaystyle \sum _{j=1}^{n_i}}{X}_{i,j},i=1,2,\dots ,k,\end{array}$$

and

$$\begin{array}{c}S(n_1,n_2,\dots ,n_k)={\displaystyle \sum _{i=1}^k}{S}_{n_i}={\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{j=1}^{n_i}}{X}_{ij},\end{array}$$

found in Wang and Wang (2007), we formulate the following result:

Let $\{X_{i,j},j\ge 1\}$ be i.i.d. non-negative random variables with common distribution function $F_i(x)$ and finite mean ${\mu }_i$ for any $i=1,\dots ,k$ and let $n_i$, for any $i=1,\dots ,k$ be a sequence of integers. Let us assume that $\{X_{i,j},j\ge 1\}$, $i=1,\dots ,k$ are mutually independent. If the distributions $F_i$ are consistently varying ($F_i\in \mathcal{C}$, for the definition see below) for any $i=1,\dots ,k$ then for any $\gamma >0$

$$\begin{array}{c}\mathbf{P}\left[S(n_1,n_2,\dots ,n_k)-{\displaystyle \sum _{i=1}^k}{n}_i{\mu }_i>x\right]\sim {\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x),\end{array}$$

holds, as $n_i\to \infty$, for any $i=1,\dots ,k$, uniformly for all $x\ge max\{\gamma n_1,\gamma n_2,\dots ,\gamma n_k\}$.

2. Preliminary Concepts

In this paper some sequence $\{X_n,n\ge 1\}$ of i.i.d. r.v.’s is considered, which represent claims in a risk model with common distribution function F and finite mean $\mu$. Let us suppose that this sequence is independent from the integer counting process $\{N(t),t\ge 0\}$, representing the claim arrival process and denote by $\lambda (t)=EN(t)<\infty$ its mean value for any $0\le t<\infty$. We assume that $\lambda (t)\to \infty$, as $t\to \infty$. All limit relationships, unless otherwise stated, are for $n\to \infty$ or $t\to \infty$.

Let us call a distribution function F as heavy-tailed distribution, if it has no exponential moments, that means $E\left[e^{ϵX}\right]=\infty ,\forall ϵ>0$. Next, we recall some useful facts from the following subclasses of heavy tailed distributions:

A distribution function F with support on $[0,\infty )$ belongs to $\mathcal{C}$ if the following asymptotic relation holds

$$\mathcal{C}=\left\{F:\underset{y\searrow 1}{lim}\underset{x\to \infty }{lim\; inf}\frac{\overline{F}(xy)}{\overline{F}(x)}=1\right\}$$

or equivalently

$$\mathcal{C}=\left\{F:\underset{y\nearrow 1}{lim}\underset{x\to \infty }{lim\; sup}\frac{\overline{F}(xy)}{\overline{F}(x)}=1\right\}.$$

For such a distribution function F, it is said to have a consistently varying tail.

A distribution function F with support on $[0,\infty )$ belongs to $\mathcal{S}$, if the following asymptotic formulas are valid

$$\mathcal{S}=\left\{F:\underset{x\to \infty }{lim}\frac{\overline{F^{n\ast }}(x)}{\overline{F}(x)}=n,\forall n\ge 2\right\}.$$

For such a distribution function F, it is said to have a subexponential tail.

Let us denote by ${\mathcal{S}}^{\ast }$ the subclass of the subexponential distributions, which contains distributions with finite mean $\mu$ and the next limit exists

$${\mathcal{S}}^{\ast }=\left\{F:\underset{x\to \infty }{lim}{\int }_0^x\frac{\overline{F}(x-y)}{\overline{F}(x)}\overline{F}(y)=2\mu \right\}.$$

This class was firstly introduced by Klüppelberg (1988).

A distribution function F with support on $[0,\infty )$ belongs to $\mathcal{L}$ if the following asymptotic holds

$$\mathcal{L}=\left\{F:\underset{x\to \infty }{lim}\frac{\overline{F}(x-y)}{\overline{F}(x)}=1,\forall y>0\right\}.$$

In this case the distribution function F is said to have long tail. For any long tail distribution there exist an non-decreasing function $h(x)$ such that $h(x)\to \infty$ as $x\to \infty$ and the following asymptotic relation holds

$$\begin{array}{c}\overline{F}\left[x\pm h(x)\right]\sim \overline{F}(x).\end{array}$$

(1)

as $x\to \infty$ (see for example (Konstantinides 2017, Lemma 8.1)). It is well known the inclusions

$$\begin{array}{c}\hfill \mathcal{C}\subseteq {\mathcal{S}}^{\ast }\subseteq \mathcal{S}\subseteq \mathcal{L}.\end{array}$$

Remark 1.

In (Foss et al. 2011, Lemma 2.19) was established the following assertion: For any long tailed distribution F we can find an increasing function $l(x)<h(x)$ such that $l(x)\to \infty$, as $x\to \infty$, for which the following holds

$$\begin{array}{c}\overline{F}\left[x\pm l(x)\right]\sim \overline{F}(x),\end{array}$$

(2)

as $x\to \infty$. Let us denote by $l^{\leftarrow }$ the inverse function of l, which represents an increasing function and the limit relation $l^{\leftarrow }(x)\to \infty$, as $x\to \infty$, holds.

In case of distribution function with regularly varying tail, when ${lim}_{x\to \infty }\overline{F}(tx)/\overline{F}(x)=t^{-\alpha }$, for some $\alpha >0$, a possible choice of the function l is $l(x)=o(x)$. From (Denisov et al. 2008, Section 8) we obtain that for distribution with zero mean and finite variance, is possible the choice $l(x)=\sqrt{x}$. Further information related with heavy-tailed distributions can be found in Embrecht et al. (1997), Borovkov and Borovkov (2008), Foss et al. (2011), Konstantinides (2017).

Let us remind the following notations: for two positive functions $a(·)$ and $b(·)$ we write

$$\begin{array}{ccc}\hfill a(x)\lesssim b(x)& if& \underset{x\to \infty }{lim\; sup}\frac{a(x)}{b(x)}\le 1,\hfill \\ \hfill a(x)\gtrsim b(x)& if& \underset{x\to \infty }{lim\; inf}\frac{a(x)}{b(x)}\ge 1,\hfill \\ \hfill a(x)\sim b(x)& if& \underset{x\to \infty }{lim}\frac{a(x)}{b(x)}=1.\hfill \end{array}$$

In the large deviations set-up, the asymptotic relation has the form:

$$\begin{array}{c}\mathbf{P}\left[S_n>x\right]\sim n\overline{F}(x),\end{array}$$

and in the precise large deviations the corresponding asymptotic relation is the following:

$$\begin{array}{c}\mathbf{P}\left[S_n-n\mu >x\right]\sim n\overline{F}(x).\end{array}$$

Both relations, hold uniformly for any $x\ge {\alpha }_n$ where ${\alpha }_n$ represents some non-negative sequence that tends to infinity.

3. Main result

The crucial step in our approach, comes from the following result by (Deniso et al. 2010, Theorem 5). For the sake of convenience we refer its short proof.

Theorem 1.

Let $\{X_k,k\ge 1\}$ be a non-negative independent and identically distributed sequence of random variables following the common distribution function $F\in {\mathcal{S}}^{\ast }$ with finite mean $\mu >0$. Then the following asymptotic relation holds

$$\begin{array}{c}\hfill \mathbf{P}\left[S_n>x\right]\sim n\overline{F}(x),\end{array}$$

(3)

as $n\to \infty$, uniformly for any $x\ge l^{\leftarrow }(n)$.

Proof. 

The uniformity in (3) is understood in the following sense

$$\underset{x\to \infty }{lim}\underset{n\le l(x)}{sup}\left|{\displaystyle \frac{\mathbf{P}\left[S_n>x\right]}{n\overline{F}(x)}}-1\right|=0.$$

that means $\forall ϵ>0$ and $\forall n\le l(x)$ there exists some $x(ϵ)<\infty$ such that, if $x>x(ϵ)$ the inequality

$$\left|{\displaystyle \frac{\mathbf{P}\left[S_n>x\right])}{n\overline{F}(x)}}-1\right|<ϵ,$$

holds, or equivalently $\forall ϵ>0$ and $\forall x\ge l^{\leftarrow }(n)$ there exists some $x^{\prime }(ϵ)=\lfloor x(ϵ)\rfloor$, where $\lfloor \alpha \rfloor$ denotes the integer part of $\alpha$, such that for any $n=\lfloor x\rfloor >x^{\prime }(ϵ)$, the inequality

$$\left|{\displaystyle \frac{\mathbf{P}\left[S_n>x\right]}{n\overline{F}(x)}}-1\right|<ϵ$$

holds and so by the arbitrariness of the choice of $\epsilon >0$ we find

$$\underset{n\to \infty }{lim}\underset{x\ge l^{\leftarrow }(n)}{sup}\left|{\displaystyle \frac{\mathbf{P}\left[S_n>x\right]}{n\overline{F}(x)}}-1\right|=0.$$

 ☐

Now, we can examine the asymptotic relation of precise large deviations for a distribution $F\in {\mathcal{S}}^{\ast }$.

Theorem 2.

Let $\{X_k,k\ge 1\}$ be a non-negative, independent and identically distributed sequence of random variables with common distribution function $F\in {\mathcal{S}}^{\ast }$ with finite mean $\mu >1$. Then holds

$$\begin{array}{c}\mathbf{P}\left[S_n-n\mu >x\right]\sim n\overline{F}(x),\end{array}$$

(4)

uniformly for $x\ge l^{\leftarrow }(n\mu )$.

Proof. 

From relation (3) we get

$$\begin{array}{c}\mathbf{P}\left[S_n-n\mu >x\right]<\mathbf{P}\left[S_n>x\right]\sim n\overline{F}(x).\end{array}$$

On the other hand for $x\ge l^{\leftarrow }(n\mu )$ and from relations (3) and (1) we obtain

$$\begin{array}{c}\mathbf{P}\left[S_n-n\mu >x\right)\sim n\overline{F}(x+n\mu ]>n\overline{F}[x+l(x)]\sim n\overline{F}(x).\end{array}$$

 ☐

The precise large deviations refers to a random walk of the type

$$\begin{array}{c}{S}_{N(t)}={\displaystyle \sum _{k=1}^{N(t)}}{X}_k,t>0\end{array}$$

(5)

where the asymptotic relation of precise large deviations is formulated as

$$\begin{array}{c}\mathbf{P}[S_{N(t)}-\mu \lambda (t)>x]\sim \lambda (t)\overline{F}(x),\end{array}$$

(6)

as $t\to \infty$, uniformly for $x\ge f(t)$, with $f(t)$ representing a non-negative function, that tends to infinity.

Therefore, Let us consider the asymptotic relation (6) for random sums, when $F\in S^{\ast }$ under the following conditions on $N(t)$:

• $\mathbf{Assumption}{\mathbf{N}}_{\mathbf{1}}$:

  $$\frac{N(t)}{\lambda (t)}\stackrel{P}{\to }1.$$
• $\mathbf{Assumption}{\mathbf{N}}_{\mathbf{2}}$: For any $ϵ>0$ and for any $\delta >0$ the following asymptotic relation holds

  $$\sum _{k>(1+\delta )\lambda (t)}{(1+ϵ)}^k\mathbf{P}[N(t)=k]=o(1).$$

Remark 2.

From Klüppelberg (1988) we see that Assumption $N_1$ permits the following equivalent formulation. There exists some positive function $ϵ(t)$, with $ϵ(t)\to 0$, such that

$$\begin{array}{c}\mathbf{P}\left[|N(t)-\lambda (t)|\le ϵ(t)\lambda (t)\right]=o(1).\end{array}$$

Theorem 3.

Let $\{X_k,k\ge 1\}$ be a non-negative, independent and identically distributed sequence of random variables with common distribution function $F\in {\mathcal{S}}^{\ast }$ with finite mean $\mu >1$. Let $\{N(t),t\ge 0\}$ be a non-negative and integer valued counting process. We assume that $\{X_k,k\ge 1\}$ and $\{N(t),t\ge 0\}$ are mutually independent. If $N(t)$ satisfies both assumptions $N_1$ and $N_2$ then the following asymptotic relation holds

$$\begin{array}{c}\mathbf{P}[S_{N(t)}-\mu \lambda (t)>x]\sim \lambda (t)\overline{F}(x),\end{array}$$

(7)

as $t\to \infty$, uniformly for any $x\ge l^{\leftarrow }\left[\mu \lambda (t)\right]$.

Proof. 

Let us use the decomposition, proposed in the proof of (Klüppelberg and Mikosch 1997, Theorem 3.1). We can state

$$\mathbf{P}[S_{N(t)}-\mu \lambda (t)>x]={\displaystyle \sum _{k=1}^{\infty }}\mathbf{P}[S_k>x+\mu \lambda (t)]\mathbf{P}[N(t)=k].$$

Further, let us split the sum in three parts

$$\begin{array}{c}=\sum _{k<(1-\delta )\lambda (t)}+\sum _{(1-\delta )\lambda (t)\le k\le (1+\delta )\lambda (t)}+\sum _{k>(1+\delta )\lambda (t)}:=I_1+I_2+I_3.\end{array}$$

(8)

Now, we can see that

$$\begin{array}{ccc}\hfill I_1& =& \sum _{k<(1-\delta )\lambda (t)}\mathbf{P}[S_k>x+\mu \lambda (t)]\mathbf{P}[N(t)=k]\hfill \\ & \le & \mathbf{P}\left[S_{(1-\delta )\lambda (t)}>x+\mu \lambda (t)\right]\sum _{k<(1-\delta )\lambda (t)}P[N(t)=k].\hfill \end{array}$$

Let us observe that $l[x+\mu \lambda (t)]\ge l(x)\ge \mu \lambda (t)\ge (1-\delta )\lambda (t)$, hence using relation (3) and (2) we obtain

$$\begin{array}{ccc}\hfill I_1& \lesssim & (1-\delta )\lambda (t)\overline{F}[x+\mu \lambda (t)]\mathbf{P}[N(t)<(1-\delta )\lambda (t]\hfill \\ & \lesssim & (1-\delta )\lambda (t)\overline{F}(x)\mathbf{P}\left[N(t)-\lambda (t)<-\delta \lambda (t)\right],\hfill \end{array}$$

so by Assumption $N_1$ and taking into account the Remark 2 it follows

$$\begin{array}{c}{I}_1=o\left[\lambda (t)\overline{F}(x)\right],\end{array}$$

(9)

as $x\ge l^{\leftarrow }(\mu \lambda (t))$.

Next, we deal with term $I_2$. Let us write

$$\begin{array}{ccc}\hfill I_2& =& \sum _{(1-\delta )\lambda (t)\le k\le (1+\delta )\lambda (t)}\mathbf{P}[S_k>x+\mu \lambda (t)]\mathbf{P}[N(t)=k]\hfill \\ & \ge & \mathbf{P}[S_{(1-\delta )\lambda (t)}>x+\mu \lambda (t)]P\left[1-\delta \le \frac{N(t)}{\lambda (t)}\le 1+\delta \right],\hfill \end{array}$$

and by Assumption $N_1$ follows

$$\begin{array}{c}{I}_2\gtrsim \mathbf{P}\left[S_{(1-\delta )\lambda (t)}>x+\mu \lambda (t)\right],\end{array}$$

since holds the inequality $x\ge l^{\leftarrow }(\mu \lambda (t))$. Further by relation (3) we find

$$\begin{array}{c}\hfill I_2\gtrsim (1-\delta )\lambda (t)\overline{F}[x+\mu \lambda (t)]>(1-\delta )\lambda (t)\overline{F}[x+l(x)].\end{array}$$

Thus, relation (1) implies

$$\begin{array}{c}\hfill I_2\gtrsim (1-\delta )\lambda (t)\overline{F}(x).\end{array}$$

Similarly we find the upper bound

$$\begin{array}{ccc}\hfill I_2& =& \sum _{(1+\delta )\lambda (t)\le k\le (1+\delta )\lambda (t)}\mathbf{P}[S_k>x]\mathbf{P}[N(t)=k]\hfill \\ & \le & \mathbf{P}\left[S_{(1+\delta )\lambda (t)}>x+\mu \lambda (t)\right]\mathbf{P}\left[1-\delta \le \frac{N(t)}{\lambda (t)}\le 1+\delta \right],\hfill \end{array}$$

and Assumption $N_1$ implies

$$\begin{array}{c}{I}_2\lesssim \mathbf{P}\left[S_{(1+\delta )\lambda (t)}>x+\mu \lambda (t)\right],\end{array}$$

as $t\to \infty$. From relations (3) and (2) and for small enough $\delta >0$ we obtain the inequality $l[x+\mu \lambda (t)]\ge (1+\delta )\lambda (t)$ and hence we find

$$\begin{array}{c}{I}_2\lesssim (1+\delta )\lambda (t)\overline{F}(x).\end{array}$$

Letting $\delta \to 0$ we get

$$\begin{array}{c}{I}_2\sim \lambda (t)\overline{F}(x),\end{array}$$

(10)

as $t\to \infty$, for any $x\ge l^{\leftarrow }\left[\mu \lambda (t)\right]$.

At last, on

$$\begin{array}{ccc}\hfill I_3& =& \sum _{k>(1+\delta )\lambda (t)}\mathbf{P}[S_k>x+\mu \lambda (t)]\mathbf{P}[N(t)=k],\hfill \end{array}$$

we apply Kesten’s inequality (see for example (Konstantinides 2017, Theorem 6.2)) for $F\in \mathcal{S}$. For any $ϵ>0$ there exists some constant $K=K(ϵ)$ such that

$$\begin{array}{c}\hfill \mathbf{P}[S_n>x]\le K{(1+ϵ)}^n\overline{F}(x),\end{array}$$

hence, we obtain

$$\begin{array}{ccc}\hfill I_3& \le & \sum _{k>(1+\delta )\lambda (t)}K{(1+ϵ)}^n\overline{F}(x)\mathbf{P}[N(t)=k],\hfill \end{array}$$

and by Assumption $N_2$ we get

$$\begin{array}{c}{I}_3\sim o\left[\lambda (t)\overline{F}(x)\right]\end{array}$$

(11)

as $t\to \infty$, uniformly for any $x\ge l^{\leftarrow }(\mu \lambda (t)).$

Putting relations (9)–(11) in (8) we find the final result. ☐

Remark 3.

Lemma 2.1 from Klüppelberg and Mikosch (1997) implies that the Poisson counting process satisfies both Assumptions $N_1$ and $N_2$. Also, the renewal counting process satisfies both Assumptions $N_1$ and $N_2$ under certain condition.

4. The Multi-Risk Model

Recent works on the multi-risk model are found in Wang and Wang (2013), Liu (2010) and Wang et al. (2014). We examine these results for the case $X_i\in S^{\ast }$.

Theorem 4.

Let $\{X_{i,j},j\ge 1\}$ be i.i.d., non-negative random variables with common distribution function $F_i(x)$ and finite mean ${\mu }_i$, for $i=1,2,\dots ,k$. Let $n_i,i=1,2,\dots ,k$ be a positive integer sequence and assume $\{X_{i,j},j\ge 1\}$, $i=1,2,\dots ,k$ are independent of $\{n_i\}$. If $F_i\in S^{\ast }$ for any $i=1,2,\dots ,k$ then the asymptotic relation

$$\begin{array}{c}\hfill \mathbf{P}\left[S(n_1,n_2,\dots ,n_k)-{\displaystyle \sum _{i=1}^k}{n}_i{\mu }_i\right]\sim {\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x)\end{array}$$

(12)

holds, as $n_i\to \infty$, for $i=1,2,\dots ,k$, uniformly for all x satisfying the inequality $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2),\dots ,{l_k}^{\leftarrow }(n_k{\mu }_k)\}$.

Proof. 

Let us employ mathematical induction: In case $k=2$ we can see:

$$\begin{array}{c}\hfill \mathbf{P}\left[S(n_1,n_2)-n_1{\mu }_1-n_2{\mu }_2>x\right]=\mathbf{P}\left[S_{n_1}+S_{n_2}-n_1{\mu }_1-n_2{\mu }_2>x\right]\\ \hfill \ge \mathbf{P}[\{S_{n_1}-n_1{\mu }_1>x+l_1(x),S_{n_2}-n_2{\mu }_2>-l_1(x)\}\\ \hfill \cup \{S_{n_2}-n_2{\mu }_2>x+l_2(x),S_{n_1}-n_1{\mu }_1>-l_2(x)\}]\\ \hfill \ge \mathbf{P}[S_{n_1}-n_1{\mu }_1>x+l_1(x)]\mathbf{P}[S_{n_2}-n_2{\mu }_2>-l_1(x)]\\ \hfill +\mathbf{P}[S_{n_2}-n_2{\mu }_2>x+l_2(x)]\mathbf{P}[S_{n_1}-n_1{\mu }_1>-l_2(x)]\\ \hfill -\mathbf{P}[S_{n_1}-n_1{\mu }_1>x+l_1(x)]\mathbf{P}[S_{n_2}-n_2{\mu }_2>x+l_2(x)],\end{array}$$

and relations (4) and (1) imply the asymptotic relations

$$\begin{array}{c}\mathbf{P}[S_{n_1}-n_1{\mu }_1>x+l_1(x)]\sim n_1{\overline{F}}_1(x),\end{array}$$

and

$$\begin{array}{c}\mathbf{P}\left[S_{n_2}-n_2{\mu }_2>x+l_2(x)\right]\sim n_2{\overline{F}}_2(x).\end{array}$$

Further, through the strong law of large numbers we find

$$\begin{array}{c}\mathbf{P}[S_{n_1}-n_1{\mu }_1>-l_2(x)]=1,\end{array}$$

and

$$\begin{array}{c}\mathbf{P}[S_{n_2}-n_2{\mu }_2>-l_1(x)]=1,\end{array}$$

so we obtain the asymptotic inequalities

$$\begin{array}{ccc}\hfill \mathbf{P}[S(n_1,n_2)>x]& \gtrsim & n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)-n_1{\overline{F}}_1(x)n_2{\overline{F}}_2(x)\hfill \\ & \gtrsim & n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)-o[n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)],\hfill \end{array}$$

(13)

as $n\to \infty$, uniformly for all $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2)\}$.

On the other hand for any fixed $0<ϵ<1/2$ we find

$$\begin{array}{c}\mathbf{P}\left[S(n_1,n_2)-n_1{\mu }_1-n_2{\mu }_2>x\right]\le \mathbf{P}\left[S_{n_1}+S_{n_2}>x\right]\hfill \\ \le \mathbf{P}\left[\{S_{n_1}>a_1\}\cup \{S_{n_2}>a_2\}\cup \{S_{n_1}>ϵx-l_1(ϵx),S_{n_2}>ϵx-l_2(ϵx)\}\right]\hfill \\ \le \mathbf{P}[S_{n_1}>a_1]+\mathbf{P}[S_{n_2}>a_2]+\mathbf{P}[S_{n_1}>ϵx-l_1(ϵx)]\mathbf{P}[S_{n_2}>ϵx-l_2(ϵx)],\hfill \end{array}$$

with $a_1:=x-l_1(x),a_2:=x-l_2(x)$, and from relations (3) and $F_i\in S^{\ast }\subseteq \mathcal{L},i=1,2$ we see that the distribution of the sum $S(n_1,n_2)$ belongs to $\mathcal{L}$. Therefore we have

$$\begin{array}{l}\mathbf{P}[S(n_1,n_2)>x]\sim \mathbf{P}[S(n_1,n_2)>x+n_1{\mu }_1+n_2{\mu }_2]\lesssim n_1\mathbf{P}[X_1>x-l_1(x)]\hfill \\ +n_2\mathbf{P}[X_2>x-l_2(x)]+n_1\mathbf{P}[X_1>ϵx-l_1(ϵx)]n_2\mathbf{P}[X_2>ϵx-l_2(ϵx)]\hfill \\ \lesssim n_1{\overline{F}}_1[x-l_1(x)]+n_2{\overline{F}}_2[x-l_2(x)]+n_1{\overline{F}}_1[ϵx-l_1(ϵx)]n_2{\overline{F}}_2[ϵx-l_2(ϵx)]\hfill \\ \lesssim n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)+n_1{\overline{F}}_1(ϵx)n_2{\overline{F}}_2(ϵx).\hfill \end{array}$$

But now, we can note that the following equality

$$\begin{array}{c}\frac{n_1{\overline{F}}_1(ϵx)n_2{\overline{F}}_2(ϵx)}{n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)}=\frac{1}{\frac{1}{n_1{\overline{F}}_1(ϵx)}·\frac{{\overline{F}}_2(x)}{{\overline{F}}_2(ϵx)}+\frac{1}{n_2{\overline{F}}_2(x)}·\frac{{\overline{F}}_1(x)}{{\overline{F}}_1(ϵx)}}\end{array}$$

holds, since

$$\begin{array}{c}\frac{{\overline{F}}_i(x)}{{\overline{F}}_i(ϵx)}\le {\left(\frac{{\overline{F}}_i[x+l_i(x)]}{{\overline{F}}_i(x)}\right)}^{-1}\sim 1,\end{array}$$

and $n_i{\overline{F}}_i(ϵx)\to 0$ as $n_i\to \infty i=1,2$, for all $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2)\}$. Hence, we conclude

$$\begin{array}{c}\hfill \mathbf{P}[S(n_1,n_2)-n_1{\mu }_1-n_2{\mu }_2>x]\\ \hfill \lesssim n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)+o[n_1{\overline{F}}_1(x)+n_2{\overline{F}}_2(x)],\end{array}$$

(14)

uniformly for all $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2)\}$. Relations (13) and (14) show that (12) holds for $k=2$.

Next, wet suppose that (12) holds for some $k-1$ and we show that then it is valid for k too.

$$\begin{array}{c}\mathbf{P}\left[S(n_1,n_2,\dots ,n_k)-{\displaystyle \sum _{i=1}^k}{n}_i{\mu }_i>x\right]\hfill \\ \ge \mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}-{\displaystyle \sum _{i=1}^{k-1}}{n}_i{\mu }_i>x+\underset{1\le i\le k-1}{min}\{l_i(x)\}\right]\hfill \\ \mathbf{P}\left[S_{n_k}-n_k{\mu }_k>-\underset{1\le i\le k-1}{min}\{l_i(x)\}\right]\hfill \\ +\mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}-{\displaystyle \sum _{i=1}^{k-1}}{n}_i{\mu }_i>x-l_k(x)\right]\mathbf{P}\left[S_{n_k}-n_k{\mu }_k>x+l_k(x)\right]\hfill \\ -\mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}-{\displaystyle \sum _{i=1}^{k-1}}{n}_i{\mu }_i>x+\underset{1\le i\le k-1}{min}\{l_i(x)\}\right]\hfill \\ \times \mathbf{P}\left[S_{n_k}-n_k{\mu }_k>x+l_k(k)\right].\hfill \end{array}$$

From the induction assumption and relations (4) and (1) follow the asymptotic relations

$$\begin{array}{c}\hfill \mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}-{\displaystyle \sum _{i=1}^{k-1}}{n}_i{\mu }_i>x+\underset{1\le i\le k-1}{min}\{l_i(x)\}\right]\sim \\ \hfill {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i\left[x+\underset{1\le i\le k-1}{min}\{l_i(x)\}\right]\gtrsim {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i[x+l_i(x)]\sim {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i(x),\end{array}$$

and

$$\begin{array}{c}\mathbf{P}\left[S_{n_k}-n_k{\mu }_k>x+l_k(x)\right]\sim n_k{\overline{F}}_k(x).\end{array}$$

Hence, the following asymptotic inequality

$$\begin{array}{c}\mathbf{P}\left[S(n_1,n_2,\dots ,n_k)-{\displaystyle \sum _{i=1}^k}{n}_i{\mu }_i>x\right]\gtrsim {\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x)-o\left({\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x)\right),\end{array}$$

(15)

holds, uniformly for any $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2),\dots ,{l_k}^{\leftarrow }(n_k{\mu }_k)\}$.

On the other hand for any $0<ϵ<\frac{1}{2}$ we obtain

$$\begin{array}{c}\hfill \mathbf{P}\left[S(n_1,n_2,\dots ,n_k)-{\displaystyle \sum _{i=1}^k}{n}_i{\mu }_i>x\right]\le \mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_k}>x\right]\\ \hfill \le \mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}>x-\underset{1\le i\le k-1}{max}\{l_i(x)\}\right]+\mathbf{P}\left[S_{n_k}>x-l_k(x)\right]\\ \hfill +\mathbf{P}\left[S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}>ϵx-\underset{1\le i\le k-1}{max}\{l_i(ϵx)\}\right]\mathbf{P}\left[S_{n_k}>ϵx-l_k(ϵx)\right].\end{array}$$

Now, by the induction assumption and through relations (3) and (1) we get

$$\begin{array}{c}\hfill \mathbf{P}\left(S_{n_1}+S_{n_2}+\dots +S_{n_{k-1}}>x-\underset{1\le i\le k-1}{max}\{l_i(x)\}\right)\\ \hfill \sim {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i\left(x-\underset{1\le i\le k-1}{max}\{l_i(x)\}\right)\gtrsim {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i[x-l_i(x)]\sim {\displaystyle \sum _{i=1}^{k-1}}{n}_i{\overline{F}}_i(x).\end{array}$$

Thus we find the inverse asymptotic relation

$$\begin{array}{ccc}\hfill \mathbf{P}[S(n_1,n_2,\dots ,n_k)>x]& \lesssim & {\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x)+o\left({\displaystyle \sum _{i=1}^k}{n}_i{\overline{F}}_i(x)\right),\hfill \end{array}$$

(16)

as $n_i\to \infty$, for $i=1,2,\dots ,k$ uniformly for $x\ge max\{l_1^{\leftarrow }(n_1{\mu }_1),l_2^{\leftarrow }(n_2{\mu }_2),\dots ,{l_k}^{\leftarrow }(n_k{\mu }_k)\}$. ☐

Further, we show the corresponding asymptotic relation for the case of random sums. Let us recall the notation

$$\begin{array}{c}{S}_{N_i(t)}={\displaystyle \sum _{j=1}^{N_i(t)}}{X}_{ij},i=1,2,3,\dots ,k,\end{array}$$

and

$$\begin{array}{c}S(k,t)={\displaystyle \sum _{i=1}^k}{S}_{N_i(t)}.\end{array}$$

Corollary 1.

Let $\{X_{i,j},j\ge 1\}$, $i=1,2,\dots ,k$ be i.i.d. non-negative random variables with common distribution function $F_i(x)$ and finite mean ${\mu }_i$. Let $N_i(t)$ be a sequence of stochastic processes and assume that $\{X_{i,j},j\ge 1\}$ and $N_i(t)$$i=1,2,\dots ,k$ are mutually independent. If $F_i\in S^{\ast }$ and $N_i(t)$ satisfy the assumptions $N_1$ and $N_2$ for all $i=1,2,\dots ,k$ then holds

$$\begin{array}{c}\mathbf{P}\left[S(k,t)-{\displaystyle \sum _{i=1}^k}{\mu }_i{\lambda }_i(t)\right]\sim {\displaystyle \sum _{i=1}^k}{\lambda }_i(t){\overline{F}}_i(x)\end{array}$$

(17)

as $t\to \infty$, uniformly for any $x\ge max\{l_1^{\leftarrow }\left[{\mu }_1{\lambda }_1(t)\right],l_2^{\leftarrow }\left[{\mu }_2{\lambda }_2(t)\right],\dots ,{l_k}^{\leftarrow }\left[{\mu }_k{\lambda }_k(t)\right]\}.$

Proof. 

We establish relation (17) by using Theorem 3 and employing mathematical induction as was done in the Theorem 4. In case $k=2$ we see that

$$\begin{array}{c}\hfill \mathbf{P}\left[S_{N_1(t)}+S_{N_2(t)}-{\mu }_1{\lambda }_1(t)-{\mu }_2{\lambda }_2(t)>x\right]\\ \hfill \ge \mathbf{P}[\{S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>x+l_1(x),S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>-l_1(x)\}\cup \\ \hfill \{S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>x+l_2(x),S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>-l_2(x)\}]\\ \hfill \ge \mathbf{P}\left[S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>x+l_1(x)\right]\mathbf{P}\left[S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>-l_1(x)\right]\\ \hfill +\mathbf{P}\left[S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>x+l_2(x)\right]\mathbf{P}\left[S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>-l_2(x)\right]\\ \hfill -\mathbf{P}\left[S_{N_1(t)}>x+l_1(x)\right]\mathbf{P}\left[S_{N_2(t)}>x+l_2(x)\right].\end{array}$$

From relations (7) and (1) follows the asymptotics

$$\begin{array}{c}\mathbf{P}\left[S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>x+l_1(x)\right]\sim {\lambda }_1(t){\overline{F}}_1(x),\end{array}$$

and

$$\begin{array}{c}\mathbf{P}\left[S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>x+l_1(x)\right]\sim {\lambda }_2(t){\overline{F}}_2(x).\end{array}$$

Hence, by the strong law of large numbers for random sums we find that

$$\begin{array}{c}\mathbf{P}\left[S_{N_1(t)}-{\mu }_1{\lambda }_1(t)>-l_2(x)\right]=1,\end{array}$$

and

$$\begin{array}{c}\mathbf{P}\left[S_{N_2(t)}-{\mu }_2{\lambda }_2(t)>-l_1(x)\right]=1.\end{array}$$

So we obtain

$$\begin{array}{c}\hfill \mathbf{P}\left[S_{N_1(t)}+S_{N_2(t)}-{\mu }_1{\lambda }_1(t)-{\mu }_2{\lambda }_2(t)>x\right]\gtrsim \\ \hfill {\lambda }_1(t){\overline{F}}_1(x)+{\lambda }_2(t){\overline{F}}_2(x)-o[{\lambda }_1(t){\overline{F}}_1(x)+{\lambda }_2(t){\overline{F}}_2(x)]\end{array}$$

(18)

as $t\to \infty$, uniformly for any $x\ge max\{l_1^{\leftarrow }\left[{\mu }_1{\lambda }_1(t)\right],l_2^{\leftarrow }\left[{\mu }_2{\lambda }_2(t)\right]\}$. On the other hand, for any $0<ϵ<1/2$ the following inequality holds

$$\begin{array}{c}\mathbf{P}[S_{N_1(t)}+S_{N_2(t)}-{\mu }_1{\lambda }_1-{\mu }_2{\lambda }_2>x]\hfill \\ \le \{\mathbf{P}[S_{N_1(t)}>x-l_1(x)]+\mathbf{P}[S_{N_2(t)}>x-l_2(x)]\hfill \\ +\mathbf{P}[S_{N_1(t)}>ϵx+l_1(ϵx)]\mathbf{P}[S_{N_2(t)}>ϵx-l_2(ϵx)].\hfill \end{array}$$

By relations (7) and (1) we find

$$\begin{array}{c}\mathbf{P}(S_{N_1(t)}+S_{N_2(t)}>x)\lesssim \hfill \\ {\lambda }_1(t){\overline{F}}_1(x)+{\lambda }_2(t){\overline{F}}_2(x)+o[{\lambda }_1(t){\overline{F}}_1(x)+{\lambda }_2(t){\overline{F}}_2(x)]\hfill \end{array}$$

(19)

as $t\to \infty$, uniformly for any $x\ge max\{l_1^{\leftarrow }\left[{\mu }_1{\lambda }_1(t)\right],l_2^{\leftarrow }\left[{\mu }_2{\lambda }_2(t)\right]\}$. From (18) and (19) we get the required result (17) for the case $k=2$. Next, we continue the induction following the same lines from the proof of Theorem 4. ☐