Motivated by the recent work of [1], we investigate the tail probability of the non-random sums in a multi-risk model

$\begin{equation} \sum\limits^{k}_{i=1}\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}. \end{equation}$

(1.1)

Here $\{X_{ij},j\geq 1\}^{k}_{i=1}$ are the sequences of upper tail asymptotic independent random variables with long tailed, and $\{C_{ij},j\geq 1\}^{k}_{i=1}$ be another nonnegative real sequences. The corresponding random sums of (1.1) is

$\begin{equation} \sum\limits^{k}_{i=1}\sum\limits^{N_{i}(t)}_{j=1}C_{ij}X_{ij}, \end{equation}$

(1.2)

where $\{N_{i}(t)\}$ be non-negative integer-value process with $\lambda_{i}(t)=N_{i}(t)$, while $\{N_{i}(t), i=1,2,\cdots, k\}$ and $\{X_{ij},j\geq 1\}^{k}_{i=1}$ are mutually independent.

Since asymptotic behavior of precise large deviations for non-random sums and random sums of random variables has important theoretical significance and extensive applications, it attracts much attention and there appears to be a lot of research literature. For recent works of this aspect, we refer the reader to [1-13]. Among these papers, [11] studied the asymptotic lower bounds of precise large deviations for sums of nonnegative and independent and identically distributed random variable sequence $\{X_{j},j\geq 1\}$. [9] extended the results of [11] to nonnegative and negatively dependent random variables. Also, [1] extended those of [11] to a multi-risk model. We will extend and improve their results to the upper tail asymptotic independent structure.

At the end of this section, we introduce some corresponding notations and concepts of this paper required. Denoted by $F(x)=P(X\leq x)$, $\overline{F}(x)=1-F(x)$, and $\lfloor x \rfloor $ is the integer part of $x$. We use the following notations for two positive functions $a_{1}(x)$ and $a_{2}(x)$

$\begin{equation} a_{1}(x)\gtrsim a_{2}(x) {\rm if} \liminf\limits_{x\rightarrow\infty}\frac{a_{1}(x)}{a_{2}(x)}\geq 1; a_{1}(x)\sim a_{2}(x) {\rm if} \lim\limits_{x\rightarrow\infty}\frac{a_{1}(x)}{a_{2}(x)}=1. \end{equation}$

Definition 1.1 [11] we say that a distribution $F$ on $(-\infty, +\infty)$ belongs to the long-tailed distribution class, denoted by$F\in \mathcal{L}$, if for any $y\in(-\infty,+\infty)$,

$\begin{equation} \overline{F}(x+y)\sim \overline{F}(x). \end{equation}$

Remark It is known that the long-tailed distribution class is one of the most important heavy-tailed distribution classes, where we say $X$ (or its distribution $F$) is heavy tailed if it has no exponential moments.Also, one can see that, a distribution $F\in \mathcal{L}$ if and only if there exists a function $h(\cdot):[0,\infty)\mapsto [0,\infty)$ such that $h(x)\rightarrow \infty$, $\lim\limits_{x\rightarrow\infty}\frac{h(x)}{x}=0$ and

$\begin{equation} \overline{F}(x+y)\sim \overline{F}(x) \end{equation}$

(1.3)

holds uniformly for all $|y|\leq h(x)$.

Definition 1.2 [12] we say that random variable sequence $\{X_{j},j\geq 1\}$ is upper tail asymptotic independent (UTAI), if all natural numbers $i\neq j$,

$ \lim\limits_{\min\{x_{i},x_{j}\}\rightarrow\infty}P\left(X_{i}>x_{i}|X_{j}>x_{j}\right)=0. $

Remark For research on this structure, we refer the reader to [12], which presented some examples to illustrate that this structure is wider than the other dependent structures.

Theorem 2.1 For $i=1,2,\cdots ,k$, let $\{X_{ij},j\geq 1\}$ be a sequence of UTAI and nonnegative random variables with common distribution $F_{i}\in \mathcal{L}$, and let $\{n_{i}\}$ be a positive integer sequence. We assume that $\{X_{ij},j\geq 1\}_{i=1}^{k}$ are mutually independent. Then, for any fixed $0< C_{ij}<\infty$, $i=1,2,\cdots ,k$, $j\geq 1$,

$\begin{equation} P(\sum\limits^{k}_{i=1}\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>x)\gtrsim \sum\limits^{k}_{i=1}\sum\limits^{n_{i}}_{j=1}P(C_{ij}X_{ij}>x)\\ \end{equation}$

(2.1)

holds $\mbox{as}\ x\rightarrow\infty$.

Proof We use induction to prove $(2.1)$.

(I) When $k=1$, for an $h(x)$ satisfying $(1.3)$, we have

$\begin{eqnarray} &&P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>x)\geq P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>x, \bigcup^{n_{1}}_{j=1}(C_{1l}X_{1l}>x+h(x)))\nonumber\\ &\geq& \sum\limits^{n_{1}}_{l=1}P(C_{1l}X_{1l}>x+h(x))-\sum\limits_{1\leq l< j\leq n_{1}}P(C_{1j}X_{1j}>x+h(x), C_{1l}X_{1l}>x+h(x))\nonumber\\ &=&\sum\limits^{n_{1}}_{l=1}P(C_{1l}X_{1l}>x+h(x))\nonumber\\ &&-\sum\limits_{1\leq l< j\leq n_{1}}P(C_{1j}X_{1j}>x+h(x)|C_{1l}X_{1l}>x+h(x))P(C_{1l}X_{1l}>x+h(x))\nonumber\\ &=&K_{1}-K_{2}. \end{eqnarray}$

(2.2)

By $F_{1}\in \mathcal{L}$, we find

$\begin{eqnarray} \liminf\limits_{x\rightarrow\infty}\frac{K_{1}}{P(\sum\limits^{n_{1}}\limits_{j=1}C_{1j}X_{1j}>x)}= \liminf\limits_{x\rightarrow\infty}\frac{P(\sum\limits^{n_{1}}\limits_{j=1}X_{1j}>\frac{x}{C_{1j}}+ \frac{h(x)}{C_{1j}})}{P(\sum\limits^{n_{1}}\limits_{j=1}C_{1j}X_{1j}>x)}\geq 1. \end{eqnarray}$

(2.3)

For $K_{2}$, along with UTAI property, we have that

$\begin{eqnarray} &&\limsup\limits_{x\rightarrow\infty}\frac{K_{2}}{P\left(\sum\limits^{n_{1}}\limits_{j=1}C_{1j}X_{1j}>x\right)}\nonumber\\ &&= \limsup\limits_{x\rightarrow\infty}\left[\sum\limits_{1\leq l< j\leq n_{1}}P(C_{1j}X_{1j}>x+h(x)|C_{1l}X_{1l}>x+h(x)) \frac{P(C_{1l}X_{1l}>x+h(x))}{P(\sum\limits^{n_{1}}\limits_{j=1}C_{1j}X_{1j}>x)}\right]\nonumber\\ &&=0. \end{eqnarray}$

(2.4)

Hence, combining (2.2)-(2.4) leads to (2.1) when $k=1$, that is

$\begin{eqnarray} P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>x)\gtrsim \sum\limits^{n_{1}}_{j=1}P(C_{1j}X_{1j}>x). \end{eqnarray}$

(2.5)

(II) For the case in which $k=2$, there is

$\begin{eqnarray*} &&P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}+\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>x)\nonumber\\ &\geq &P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>x+h(x),\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>-h(x))\nonumber \\ &&+P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>-h(x),\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>x+h(x))\nonumber\end{eqnarray*}$

$\begin{eqnarray*} &=&P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>x+h(x))P(\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>-h(x))\nonumber\\ &&+P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}>-h(x))P(\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>x+h(x)).\nonumber\\ \end{eqnarray*}$

For any $0<\delta <1$, by (2.5) and $F_{i}\in \mathcal{L}$, $i=1,2$, for sufficiently large $x$, we get

$\begin{eqnarray} P(\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>x+h(x))\geq (1-\delta)[\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>x], i=1,2. \end{eqnarray}$

(2.6)

Since for $i=1,2,$ $\{X_{ij},j\geq 1\}$ are nonnegative, for any fixed $0< C_{ij}<\infty$, $j=1,2,\cdots, n_{i} $, we have

$\begin{eqnarray} P(\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>-h(x))>(1-\delta), i=1,2. \end{eqnarray}$

(2.7)

Then, using (2.6)-(2.7), we obtain

$\begin{eqnarray*} P(\sum\limits^{n_{1}}_{j=1}C_{1j}X_{1j}+\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>x) \geq (1-\delta)^{2}P[\sum\limits^{n_{1}}_{j=1}P(C_{1j}X_{1j}>x)+P(\sum\limits^{n_{2}}_{j=1}C_{2j}X_{2j}>x)]. \end{eqnarray*}$

Therefore, letting $\delta \downarrow 0$, we obtain (2.1) when the case of $k=2$.

(III) Now suppose that (2.1) holds for $k-1$. As for $k$, using a similar argument to that in (II), for any $0<\delta <1$ and any fixed $C_{ij}$, $i=1,2,\cdots , k$, $j\geq 1$, and when $x$ is sufficiently large, we have

$\begin{eqnarray*} &&P(\sum\limits^{k}_{i=1}\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>x)\nonumber\\ &\geq& P(\sum\limits^{k-1}_{i=1}\sum\limits^{n_{i}}_{j=1}C_{ij}X_{ij}>x+h(x))+P(\sum\limits^{n_{k}}_{j=1}C_{kj}X_{kj}>-h(x))\nonumber\\ &\geq &(1-\delta)^{2}[\sum\limits^{k-1}_{i=1}\sum\limits^{n_{i}}_{j=1}P(C_{ij}X_{ij}>x+h(x))]+(1-\delta)^{2}\sum\limits^{n_{k}}_{j=1}P(C_{kj}X_{kj}>-h(x))\nonumber\\ &=&(1-\delta)^{2}\sum\limits^{k}_{i=1}\sum\limits^{n_{i}}_{j=1}P(C_{ij}X_{ij}>x+h(x)).\nonumber \end{eqnarray*}$

Letting $\delta \downarrow 0$, we obtain the desired result, and the proof Theorem 2.1 is now complete.

Theorem 3.1 For $i=1,2,\cdots ,k$, let$\{X_{ij},j\geq 1\}$ be a sequence of UTAI and nonnegative random variables with common distribution $F_{i}\in \mathcal{L}$, and let $\{N_{i}(t)\}$ be non-negative integer-value process with $\lambda_{i}(t)=N_{i}(t)$. We assume that $\{X_{ij},j\geq 1\}_{i=1}^{k}$ and $\{N_{i}(t), i=1,2,\cdots ,k\}$ are mutually independent and that $\{N_{i}(t), i=1,2,\cdots ,k\}$ satisfies

$ {\rm Assumption I}: \frac{N_{i}(t)}{\lambda_{i}(t)}\rightarrow_{p}1 {\rm as}t\rightarrow \infty. $

Then, for any fixed $0< C_{ij}<\infty$, $i=1,2,\cdots ,k$, $j\geq 1$,

$\begin{equation} P(\sum\limits^{k}_{i=1}\sum\limits^{N_{i}(t)}_{j=1}C_{ij}X_{ij}>x)\gtrsim \sum\limits^{k}_{i=1}\sum\limits^{N_{i}(t)}_{j=1}P(C_{ij}X_{ij}>x) \end{equation}$

(3.1)

holds $\mbox{as}\ x\rightarrow\infty$.

Proof Again by induction, as in the proof of Theorem 3.1, it is sufficient to show that (3.1) holds for $k=1,2$.

(I) Taking $k=1$, for any $0<\delta <1$ and any fixed $C_{1j}$, $j\geq 1$, and for sufficiently large $t$,

$\begin{eqnarray*} &&P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>x)=\sum\limits^{\infty}_{n=1}P(\sum\limits^{n}_{j=1}C_{1j}X_{1j}>x)P(N_{1}(t)=n) \\ &\geq& \sum\limits_{\lfloor (1-\delta)\lambda_{1}(t)\rfloor \leq k\leq \lfloor (1+\delta)\lambda_{1}(t)\rfloor}P(\sum\limits^{n}_{j=1}C_{1j}X_{1j}>x)P(N_{1}(t)=n)\\ &\geq& P(\sum\limits^{\lfloor(1-\delta)\lambda_{1}(t)\rfloor}_{j=1}C_{1j}X_{1j}>x)P(|\frac{N_{1}(t)}{\lambda_{1}(t)}-1|<\delta )\\ &\geq& (1-\delta)^{2}\sum\limits^{\lfloor(1-\delta)\lambda_{1}(t)\rfloor}_{j=1}P(C_{1j}X_{1j}>x), \end{eqnarray*}$

where the last inequality holds due to Assumption I and (2.5). Letting $\delta \downarrow 0$, Theorem 3.1 holds for $k=1$, that is

$\begin{equation} P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>x)\gtrsim \sum\limits^{N_{1}(t)}_{j=1}P(C_{1j}X_{1j}>x). \end{equation}$

(3.2)

(II) When $k=2$, we get

$\begin{eqnarray*} &&P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}+\sum\limits^{N_{2}(t)}_{j=1}C_{2j}X_{2j}>x)\nonumber\\ &\geq& P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>x+h(x),\sum\limits^{N_{2}(t)}_{j=1}C_{2j}X_{2j}>-h(x))\nonumber\\ &&+ P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>-h(x),\sum\limits^{N_{2}(t)}_{j=1}C_{2j}X_{2j}>x+h(x))\nonumber\end{eqnarray*}$

$\begin{eqnarray*} &=&P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>x+h(x))P(\sum\limits^{N_{2}(t)}_{j=1}C_{2j}X_{2j}>-h(x))\nonumber\\ &&+P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}>-h(x))P(\sum\limits^{N_{2}(t)}_{j=1}C_{2j}X_{2j}>x+h(x)). \end{eqnarray*}$

By (3.2) and $F_{i}\in \mathcal{L}(i=1,2)$, for any $0<\delta <1$, for sufficiently large $t$, we arrive

$\begin{eqnarray} P(\sum\limits^{N_{i}(t)}_{j=1}C_{ij}X_{ij}>x+h(x))\geq (1-\delta)[P(\sum\limits^{N_{i}(t)}_{j=1}C_{ij}X_{ij}>x)], i=1,2. \end{eqnarray}$

(3.3)

Since for $i=1,2,$ $\{X_{ij},j\geq 1\}$ are nonnegative, and for any fixed $0< C_{ij}<\infty$, $j\geq1$, we have

$\begin{eqnarray} P(\sum\limits^{N_{i}(t)}_{j=1}C_{ij}X_{ij}>-h(x))>(1-\delta), i=1,2. \end{eqnarray}$

(3.4)

Then, using (3.3)-(3.4), we obtain

$\begin{eqnarray*} &&P(\sum\limits^{N_{1}(t)}_{j=1}C_{1j}X_{1j}+\sum\limits^{N_{2}(t)}_{j=1}C_{1j}X_{1j}>x)\nonumber\\ &\geq& (1-\delta)^{2}P[\sum\limits^{N_{1}(t)}_{j=1}P(C_{1j}X_{1j}>x)+P(\sum\limits^{N_{2}(t)}_{j=1}C_{1j}X_{1j}>x)]. \end{eqnarray*}$

Therefore, letting $\delta \downarrow 0$, we obtain (3.1) for $k=2$. The proof Theorem 3.1 is completed.