# Precise large deviations of some net loss processes in three nonstandard
renewal risk models

Zhaolei Cui1 and Yuebao Wang2
1 School of mathematics and statistics, Changshu Institute of Technology,
Suzhou 215000, China
2 School of Mathematical Sciences, Soochow University, Suzhou 215006, China
Research supported by National Natural Science Foundation of China (No.s
11071182).Corresponding author. Telephone: +86 512 67422726. Fax: +86 512
65112637. E-mail<EMAIL_ADDRESS>
Abstract

For three nonstandard renewal risk models, in which claim sizes are
identically distributed random variables with consistently varying tails, we
study precise large deviations of proportional net loss process and excess-of-
net-loss process under the consideration of income factor. Then we apply the
above results to obtain asymptotic estimate of mean of stop-net-loss
reinsurance treat and random-time ruin probability. In addition, we have
improved and generalized some known related results, none of which involve the
income factor.

Keywords: precise large deviation; nonstandard renewal risk model;
consistently varying tail distribution; proportional net loss process; excess-
of-net-loss process; stop-net-loss reinsurance process; random-time ruin
probability

2000 Mathematics Subject Classification: Primary 60F10; 60F05; 60G50

##  1 Preliminary

It is well known that in a risk model, the initial capital and operating time
of an insurance or reinsurance company are very large. Thus, we should
estimate the asymptotic properties of some risk objects in the case that the
two prerequisites tend to infinity together. Then, precise large deviation
naturally becomes important task of this research. And the behavior of precise
large deviation is closely related to properties of the distributions of
relevant random variables and their dependent structure. Therefore, before
giving main results of this paper, we introduce the concepts and marks of
related distribution class, dependent structure of random variables and three
nonstandard renewal risk models in this section.

In this paper, all limit relations refer to $x\to\infty$ and all distributions
are supported on $(-\infty,\infty)$ or $[0,\infty)$ without a special
statement. In addition, the following marks and conventions are used
throughout the text.

Let two functions $g_{1}(\cdot)$ and $g_{2}(\cdot)$ be positive eventually.
For $1\leq i\neq j\leq 2$, we set

$g_{i,j}=\limsup g_{i}(x)g_{j}^{-1}(x).$

Then $g_{i}(x)=O\big{(}g_{j}(x)\big{)}$ means $g_{i,j}<\infty$,
$g_{1}(x)\asymp g_{2}(x)$ means $\max\\{g_{1,2},g_{2,1}\\}<\infty$,
$g_{i}(x)\lesssim g_{j}(x)$ means $g_{i,j}\leq 1$, $g_{1}(x)\sim g_{2}(x)$
means $g_{1,2}=g_{2,1}=1$, and $g_{i}(x)=o\big{(}g_{j}(x)\big{)}$ means
$g_{i,j}=0$.

We also denote the tail of distribution $F$ by $\overline{F}=1-F$, and the
$n$-fold convolution of $F$ with itself by $F^{*n}$ for all $n\geq 2$.

Other marks and conventions will be given in succession below.

###  1.1 Some distribution classes

We say that the distribution class

$\displaystyle{\cal L}=\big{\\{}F:\overline{F}(x-t)\sim\overline{F}(x)\
\text{for each}\ t\in(-\infty,\infty)\big{\\}}$

is long-tailed, and the distribution class

$\displaystyle{\cal S}=\big{\\{}F:F\in\mathcal{L}\ \ \text{and}\ \
\overline{F^{*2}}(x)\sim 2\overline{F}(x)\big{\\}}.$

is subexponential, which was introduced by Chistyakov [8]. Further, Lemma 2 of
Chistyakov [8] notes that the requirement $F\in\mathcal{L}$ is unnecessary if
$F$ is supported on $[0,\infty)$. For a detailed description of class
$\mathcal{S}$, see Foss et al. [15], etc.

In addition, for each $y\in(0,\infty)$, we set

$\overline{F}_{*}(y)=\liminf{\overline{F}}(xy){\overline{F}}^{-1}(x),~{}~{}\overline{F}^{*}(y)=\limsup{\overline{F}}(xy){\overline{F}}^{-1}(x)\
\ \text{and}\ \ L_{F}=\lim\limits_{y\downarrow 1}\overline{F}_{*}(y).$

Then we say that, the distribution classes

$\displaystyle\mathcal{D}=\big{\\{}F:{\overline{F}}_{*}(y)>0\ \ \text{for
each}\ y\in(1,\infty)\big{\\}},\ \ {\cal C}=\\{F:L_{F}=1\\},$
$\displaystyle\mathcal{ERV}(\alpha,\beta)=\\{F:y^{-\beta}\leq\overline{F}_{*}(y)\leq\overline{F}^{*}(y)\leq
y^{-\alpha}\ \ \text{for each}\ y>1\\}\ \ \text{for each pair}\ \
0<\alpha\leq\beta<\infty$

and

$\mathcal{ERV}=\cup_{0<\alpha\leq\beta<\infty}\mathcal{ERV}(\alpha,\beta)$

are dominated varying tailed, consistently varying tailed, extended regularly
varying tailed with indexes $0<\alpha\leq\beta<\infty$ and extended regularly
varying tailed, respectively. Particularly, if $\alpha=\beta$, then
$\mathcal{ERV}(\alpha,\beta)$ reduces to the regularly varying tailed
distribution class, denoted by $\mathcal{R}_{\alpha}$. Some properties of
these distributions are introduced by the following two propositions.

######  Proposition 1.1.

The following inclusion relations are proper: for each pair
$0<\alpha\leq\beta<\infty$,

$\mathcal{R}_{\alpha}\subset\mathcal{ERV}(\alpha,\beta)\subset\mathcal{C}\subset\mathcal{L}\cap\mathcal{D}\subset\mathcal{S}\subset\mathcal{L}.$

Further, we denote the moment index of $F$ by

$I_{F}=\sup\Big{\\{}s:\int_{0}^{\infty}y^{s}F(dy)<\infty\Big{\\}},$

and the upper Matuszewska index and lower Matuszewska index of distribution
$F$ by

$\displaystyle
J_{F}^{+}=-\lim\limits_{y\to\infty}\ln\overline{F}_{*}(y)\ln^{-1}y\ \ \ \
\text{and}\ \ \ \
J_{F}^{-}=-\lim\limits_{y\to\infty}\ln\overline{F}^{*}(y)\ln^{-1}y.$

######  Proposition 1.2.

Let $F$ be a distribution in class $\mathcal{D}$.

$(i)$ $0\leq J_{F}^{-}\leq I_{F}\leq J_{F}^{+}<\infty$.

$(ii)$ For each $p\in(-\infty,J_{F}^{-})$, $\overline{F}(x)=o(x^{-p})$.

$(iii)$ For each $p\in(J_{F}^{+},\infty)$,
$x^{-p}=o\big{(}\overline{F}(x)\big{)}$.

The above concepts and properties can be found in some references, such as
Feller [16], Bingham et al. [2], Cline and Samorodnitsky [10], Embrechts et
al. [14], Klüppelberg and Mikosch [23] and Tang and Tsitsiashvili [41].

###  1.2 Some dependent structures

Wang et al. [48] introduced the concept of widely dependence structure of
random variables. By definition, $X_{i},i\geq 1$ are said to be widely upper
orthant dependent (WUOD), if for each $n\geq 1$, there exists some positive
number $g_{U,F}(n)$ such that,

$\displaystyle P\Big{(}\bigcap^{n}_{i=1}\\{X_{i}>x_{i}\\}\Big{)}\leq
g_{U,F}(n)\prod_{i=1}^{n}P(X_{i}>x_{i}),\ \ \ \ \ \ x_{i}\in(-\infty,\infty),\
\ \ 1\leq i\leq n;$ (1.1)

they are said to be widely lower orthant dependent (WLOD), if for each $n\geq
1$, there exists some positive number $g_{L,F}(n)$ such that,

$\displaystyle P\Big{(}\bigcap^{n}_{i=1}\\{X_{i}\leq x_{i}\\}\Big{)}\leq
g_{L,F}(n)\prod_{i=1}^{n}P(X_{i}\leq x_{i}),\ \ \ \ \
x_{i}\in(-\infty,\infty),\ \ \ 1\leq i\leq n;$ (1.2)

and they are said to be widely orthant dependent (WOD) if they are both WUOD
and WLOD.

WUOD, WLOD and WOD structures can be called widely dependent (WD) as a joint
name. And $g_{U,F}(n),~{}g_{L,F}(n),\ n\geq 1$, are called dominating
coefficients. Clearly, $g_{U,F}(n)\geq 1,~{}g_{L,F}(n)\geq 1,n\geq 2$ and
$g_{U,F}(1)=g_{L,F}(1)=1$.

Some basic properties of WD random variables are as follows, see Proposition
1.1 of Wang et al. [48].

######  Proposition 1.3.

$(i)$ Let $X_{i},i\geq 1$ be WLOD (or WUOD). If $f_{i}(\cdot),i\geq 1$ are
nondecreasing, then $f_{i}(X_{i}),i\geq 1$ are still WLOD (or WUOD); if
$f_{i}(\cdot),i\geq 1$ are nonincreasing, then $f_{i}(X_{i}),i\geq 1$ are WUOD
(or WLOD).

$(ii)$ If $X_{i},i\geq 1$ are nonnegative and WUOD, then for each $n\geq 1$,

$E\prod^{n}_{i=1}X_{i}\leq g_{U,F}(n)\prod^{n}_{i=1}EX_{i}.$

Therefore, if $X_{i},i\geq 1$ are real value and WUOD, then for each $n\geq 1$
and any $s>0$,

$Ee^{s\sum^{n}_{i=1}X_{i}}\leq g_{U,F}(n)\prod^{n}_{i=1}Ee^{s{X_{i}}}.$

Further, Wang et al. [48] provided some examples of WD random variables, which
showed that the WD structure may contain common negatively dependent random
variables, some positively dependent random variables, and some others. When
$g_{U,F}(n)=g_{L,F}(n)=M$ for all $n\geq 1$ and some $M\in(0,\infty)$, the
inequalities (1.1) and (1.2) describe extended negatively upper and lower
orthant dependent (ENUOD and ENLOD) random variables, respectively.
$X_{i},i\geq 1$ are said to be extended negatively orthant dependent (ENOD) if
they are both ENUOD and ENLOD. ENOD, ENUOD, ENLOD random variables are called
collectively END r.v.s, see Liu [29]. Further, if $M=1$, then we have the
corresponding notions of NUOD, NLOD, NOD and ND random variables, see Ebrahimi
and Ghosh [13] and Block et al. [3].

In particular, we say that random variables $X_{i},i\geq 1$ are negatively
associated (NA), if for any disjoint nonempty subsets $A$ and $B$ of
$\\{1,\cdots,m\\},\ m\geq 2$ and any coordinate wise nondecreasing functions
$f_{1}(\cdot)$ and $f_{2}(\cdot)$, the inequality

$\displaystyle Cov(f_{1}(X_{i}:i\in A),f_{2}(X_{j}:j\in B))\leq 0$

holds whenever the moment involved exists. See Joag-Dev and Proschan [21].

###  1.3 Nonstandard renewal risk models

Here, we introduce some concepts and properties of renewal risk model for a
insurance company.

In this model, let claim sizes $Y_{i},\ i\geq 1$ be random variables with
common distribution $G$ on $[0,\infty)$ and mean $\mu_{G}\in(0,\infty)$; the
inter-arrival times of the claim sizes $Z_{i},\ i\geq 1$ also be random
variables with common distribution $H$ on $[0,\infty)$ and mean
$\mu_{H}\in(0,\infty)$. In addition, let random time $\tau$ be a nonnegative
random variable.

The times of successive claims, $\tau_{n}=\sum_{i=1}^{n}Z_{i},\ n\geq 1$,
constitute a renewal counting process
$N(t)=\sum_{n=1}^{\infty}\textbf{1}_{\\{\tau_{n}\leq\ t\\}}$ with the
corresponding renewal function $\lambda(t)=EN(t),\ t\in[0,\infty)$. For
convenience, we might as well assume that
$\inf\\{t:\lambda(t)\in(0,\infty)\\}=0$. In addition, let $c\in(0,\infty)$ be
constant interest rate and $x\in[0,\infty)$ be initial capital of the
insurance company. In order to ensure normal operation of the insurance
company, a safe load condition is usually assumed that

$\mu_{G}<c\mu_{H}.$

The above model is called standard renewal risk model, if $(Y_{i},Z_{i}),i\geq
1$ are independent and identically distributed, and if $\\{Y_{i},\ i\geq
1\\}$, $\\{Z_{i},\ i\geq 1\\}$ and $\tau$ are independent of each other.
Otherwise, it is called nonstandard and $N(t)$ is called quasi renewal
counting process.

In this paper, some renewal theorems plays an important role for the study of
some nonstandard renewal risk models. The following three results for the
quasi counting process $N(t)$ generated by WD random variables $Z_{i},i\geq 1$
are presented as below. The first result comes from Lemma 2.2 of Wang and
Cheng [42].

######  Proposition 1.4.

In a nonstandard renewal risk model, let $Z_{i},\ i\geq 1$ be WLOD random
variables with dominating coefficients $g_{L,H}(n),\ n\geq 1$. For some
$a\in(0,\infty)$, if

$\displaystyle\lim_{n\to\infty}g_{L,H}(n)e^{-an}=0,$ (1.3)

then for each $\delta\in(0,\infty)$, there exists
$r=r(H,a,\delta)\in(0,\infty)$ such that

$\displaystyle\lim_{t\to\infty}Ee^{rN(t)}\textbf{\emph{1}}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}=0.$
(1.4)

The second result dues to Lemma 4.2 of Wang et al. [44].

######  Proposition 1.5.

In a nonstandard renewal risk model, let $Z_{i},\ i\geq 1$ be WLOD random
variables with the dominating coefficients $g_{L,H}(n),\ n\geq 1$. For some
$a\in(0,\infty)$, if

$\displaystyle\lim_{n\to\infty}g_{L,H}(n)n^{-a}=0,$ (1.5)

then for each $k\geq 1$,

$\displaystyle\lim_{t\to\infty}EN^{k}(t)(\mu_{H}^{-1}t)^{-k}=1.$ (1.6)

The final result belongs to Theorem 2.4 of Wang and Cheng [43] or Theorem 4 of
Chen et al. [4], which makes up a gap in proof of Theorem 1.4 of Wang and
Cheng [42]. In the following, we say that a positive function $g(\cdot)$ is
almost decreasing (or nonincreasing), denoted by $g(x)\
\overline{\downarrow}$, if there exist two positive finite constants
$x_{0}=x_{0}\big{(}g(\cdot)\big{)}$ and $C=C(x_{0})$ such that

$\sup_{x\in[x_{0},y]}g(y)g^{-1}(x)\leq C.$

Similarly, we say that a positive function $g(\cdot)$ is almost increasing (or
nondecreasing), denoted by $g(x)\ \underline{\uparrow}$, if $g^{-1}(\cdot)$ is
almost decreasing.

######  Proposition 1.6.

In a nonstandard renewal risk model, let $Z_{i},\ i\geq 1$ be WOD random
variables with dominating coefficients $g_{U,H}(n)$ and $g_{L,H}(n),\ n\geq
1$. If there exists some almost increasing function $g(\cdot)$, some $m\geq 1$
and $\omega\in(0,1)$ such that $EZ_{1}g(Z_{1})<\infty$,

$\displaystyle\frac{x^{m-1}}{g(x)}\ \overline{\downarrow}\ 0,\ \ \
\frac{g(x)}{x^{m-1+\omega}}\ \overline{\downarrow}\ 0\ \ \ \text{and}\ \ \
\max\big{\\{}g_{U,H}(n),g_{L,H}(n)\big{\\}}\leq g(n),\ \ \ n\geq 1,$ (1.7)

then

$\displaystyle\lim_{n\to\infty}n^{-1}\sum_{i=1}^{n}Z_{i}=\mu_{H}\ \ \ \
\text{and}\ \ \ \ \lim_{t\to\infty}N(t)\mu_{H}t^{-1}=1,\ \ \ \ a.s.$ (1.8)

Among the above three propositions, it is clear that condition (1.3) is the
weakest and condition (1.7) is the strongest. In Proposition 1.6, if
$EZ_{1}^{r}\in(0,\infty)$ for some $r\in[1,\infty)$, then
$g(x)=x^{r-1}g_{0}(x)$ for $x\in[0,\infty)$, where $g_{0}(\cdot)$ is a
positive function such that $g_{0}(x)\underline{\uparrow}\infty$ and
$g_{0}(x)x^{-\omega}\overline{\downarrow}0$ for any $\omega\in(0,1)$.
Therefore, the larger the value of $r$ is, the larger the range of $g(\cdot)$
is; when $r=1$, the range of $g(\cdot)$ is small.

In Proposition 1.6, if $Z_{i},\ i\geq 1$ are ENOD random variables, then we
only need the condition $EZ_{1}\in(0,\infty)$. At this time, there is a
positive function $g_{0}(\cdot)$ such that
$g_{0}(x)\underline{\uparrow}\infty$, $EZ_{1}g_{0}(Z_{1})\in(0,\infty)$ and
condition (1.7) is satisfied for $m=1$ and $g(\cdot)=g_{0}(\cdot)$.

The paper is organized as follows. In Section 2, we present main results of
this paper. We prove these results in Section 4. To this end, we introduce
some preliminary results on the precise large deviations for non random sums
and random sums of nonnegative and noncentralized ENOD random variables in
Section 3. Finally, we give two applications of the main results in Section 5.

##  2 Main results

In this section, we give the precise large deviations of proportional net loss
process and excess-of-net-loss process, for the following three cases of the
above nonstandard renewal risk model.

Case $i)$. $\\{Y_{i},i\geq 1\\}$, $\\{Z_{i},i\geq 1\\}$ and $\tau$ are
independent of each other.

Case $ii)$. $(Y_{i},Z_{i}),\ i\geq 1$ and $\tau$ are independent of each
other, but $(Y_{1},Z_{1})$ is allowed to be arbitrarily dependent. See Chen et
al. [6].

Case $iii)$. $\\{(Y_{i},Z_{i}),i\geq 1\\}$ and $\tau$ are independent of each
other, but $\\{Y_{i},i\geq 1\\}$ and $\\{Z_{i},i\geq 1\\}$ are allowed to be
arbitrarily dependent. See Gao et al. [17].

Clearly, Case $iii)$ is the weakest condition which includes the other two
cases.

###  2.1 Precise large deviations of the proportional net loss process in Case
$i)$

In a renewal risk model, at time $t\in[0,\infty)$, a reinsurance company
covers percent $q_{1}\cdot 100$ of the total claim amount
$S_{N(t)}^{G}=\sum_{i=1}^{N(t)}Y_{i}$ for some $q_{1}\in(0,1]$; the
reinsurance company benefits percent $q_{2}\cdot 100$ of the total income
$cS_{N(t)}^{H}=c\sum_{i=1}^{N(t)}Z_{i}$ for some $q_{2}\in[0,1]$. Then we
respectively introduce the concepts of proportional reinsurance process and
proportional net loss process for the reinsurance company defined by

$R_{11}(t)=q_{1}S_{N(t)}^{G},\ \ \ t\in[0,\infty)$

and

$R_{12}(t)=q_{1}S_{N(t)}^{G}-q_{2}cS_{N(t)}^{H}=q_{1}\sum_{i=1}^{N(t)}\big{(}Y_{i}-q_{1}^{-1}q_{2}cZ_{i}\big{)}=q_{1}\sum_{i=1}^{N(t)}X_{i}=q_{1}S_{N(t)}^{F},\
\ \ t\in[0,\infty),$

where $F$ is the distribution of $X_{1}$ and $S_{n}^{F}=\sum_{i=1}^{n}X_{i},\
n\geq 1$. Clearly, $F$ is supported on $(-\infty,\infty)$. Naturally, we
require the following more general safety load condition to be satisfied:

$q_{1}\mu_{G}<q_{2}c\mu_{H}.$

The research goal $R_{11}(t)$ comes from Example 5.2 $(i)$ of Klüppelberg and
Mikosch [23], which have obtained the following conclusion. For each
$\gamma\in(q_{1}\mu_{G},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\lambda(t),\infty)}\Big{|}P\big{(}R_{11}(t)>x\big{)}\Big{(}\lambda(t)\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\lambda(t)\big{)}\Big{)}^{-1}-1\Big{|}=0$

under the assumption that $Y_{i},\ i\geq 1$ are independent random variables
with common distribution $G\in\mathcal{ERV}(\alpha,\beta)$ for some pair
$1<\alpha\leq\beta<\infty$ and $Z_{i},\ i\geq 1$ are also independent
identically distributed random variables.

In this paper, our main research goal is $R_{12}(t)$. Clearly, $R_{12}(t)$ is
naturally inspired by $R_{11}(t)$. Usually, the former should be more
interesting and more difficult than the latter in relevant researches. And the
former has a wider application, see Theorem 5.2 below. However, we haven’t
found any literature about precise large deviation of $R_{12}(t)$.

######  Theorem 2.1.

$(i)$ In Case $i)$, let $Y_{i},i\geq 1$ be ENOD random variables with common
distribution $G\in\mathcal{C}$, and let $Z_{i},\ i\geq 1$ be WOD random
variables with dominating coefficients $g_{U,H}(n)$ and $g_{L,H}(n),\ n\geq
1$. If there exists some almost increasing function $g(\cdot)$, some $m\geq 1$
and some $\omega\in(0,1)$ such that $EZ_{1}g(Z_{1})\in(0,\infty)$ and
condition (1.7) is satisfied, then for each $\gamma\in(q_{1}\mu_{G},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\mu_{H}^{-1}t,\infty)}\Big{|}P\big{(}R_{12}(t)>x\big{)}\big{(}\mu_{H}^{-1}t\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}^{-1}-1\Big{|}=0.$
(2.1)

$(ii)$ In $(i)$, if $Z_{i},\ i\geq 1$ are only required to be WLOD random
variables and condition (1.7) is weakened to condition (1.5), then for each
$\gamma\in(q_{1}\mu_{G},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\mu_{H}^{-1}t,\infty)}\Big{|}P\big{(}R_{11}(t)>x\big{)}\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t)\big{)}^{-1}-1\Big{|}=0.$
(2.2)

######  Remark 2.1.

$(i)$ In $\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t)$ of
(2.1), $q_{1}^{-1}x,\ \mu_{G}\mu_{H}^{-1}t$ and $q_{2}cq_{1}^{-1}t$
respectively reflect the effects of proportionality, decentralization and
income $cq_{2}S_{N(t)}^{H}=cq_{2}\sum_{i=1}^{N(t)}Z_{i}$.

$(ii)$ There is no substantive difference between $\lambda(t)$ and
$\mu_{H}^{-1}t$, because when $t\to\infty$, $\lambda(t)\sim\mu_{H}^{-1}t$
according to Proposition 1.6 (thus Proposition 1.5). However, we prefer to
replace $\lambda(t)$ with $\mu_{H}^{-1}t$, which seems more intuitive and
explicit.

$(iii)$ If $X_{i}=q_{1}Y_{i}-cq_{2}Z_{i},\ i\geq 1$ are ENOD random variables
with common distribution $F$, then the mean
$\mu_{F}=q_{1}\mu_{G}-cq_{2}\mu_{H}\in(-\infty,0)$ by the safety load
condition. Therefore, the conclusion (2.1) can also be obtained according to
Theorem 1.1 of Tang [39] for NOD random variables and Theorem 2.1 of Liu [29]
for ENOD random variables under the additional conditions that

$E(X_{1}^{-})^{r}<\infty\ \ \ \text{for some}\ r\in(1,\infty)\ \ \ \text{and}\
\ \ \ F(-x)=o\big{(}\overline{F}(x)\big{)},$

where $X_{1}^{-}=-X_{1}\emph{{1}}_{\\{X_{1}<0\\}}$. In proof of Theorem 2.1
$(i)$ of our paper, however, using a method which is different from that of
[39] and [29], we can successfully avoid these additional conditions.

###  2.2 Precise large deviations of the total net loss process in Case $ii)$

Especially, when $q_{1}=q_{2}=1$, $R_{11}(t)$ is called the total claim amount
at time $t\in[0,\infty)$, see Subsection 1.3.3 of Embrechts et al. [14]. Then
$R_{12}(t)$ is called the total net loss at time $t\in[0,\infty)$. For the
sake of brevity, we only study the total net loss in the subsection 2.2 and
2.3.

Recently, Theorem 1.1 of Chen et al. [6] obtained an interesting result of
precise large deviations for the following goal

$\displaystyle
R^{*}_{11}(t)=R_{11}(t)-\mu_{G}\mu_{H}^{-1}t=\sum_{i=1}^{N(t)}Y_{i}-\mu_{G}\mu_{H}^{-1}t$

in Case $ii)$. Using the result and Theorem 2.1 of this paper, we can obtain a
corresponding result for the total net loss process $R_{12}(t)$ for
$t\in[0,\infty)$ under the same conditions as Theorem 1.1 of Chen et al. [6].

######  Theorem 2.2.

In Case $ii)$, if $q_{1}=q_{2}=1$, $G\in\mathcal{C}$ and
$\overline{H}(x)=o\big{(}\overline{G}(x)\big{)},$ then for each pair
$\mu_{G}<\gamma<\Gamma<\infty$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma t,\Gamma
t]}\Big{|}P\big{(}R_{12}(t)>x\big{)}\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct)\big{)}^{-1}-1\Big{|}.$
(2.3)

######  Remark 2.2.

In Theorem 1.1 of Chen et al. [6] and Theorem 2.2 of this paper, the range of
$\gamma$ is different. The former requires $\gamma\in(0,\infty)$ and the
latter requires $\gamma\in(\mu_{G},\infty)$. This is because the former
research goal is centralized, while the latter research goal is
noncentralized. In fact, the range $\gamma\in(0,\infty)$ for $R^{*}_{11}(t)$
is equivalent to $\gamma\in(\mu_{G},\infty)$ for $R_{11}(t)$, as well as
$R_{12}(t)$. For this, we will give a specific explanation in proof of Lemma
3.1 below.

###  2.3 Precise large deviations of the total net loss process in Case $iii)$

In Theorem 2.2, if we slightly strengthen the condition $EZ_{1}\in(0,\infty)$
to $EZ_{1}^{r}\in(0,\infty)$ for some $r\in(1,\infty)$, then can greatly
expand the application scope of the result

######  Theorem 2.3.

In Case $iii)$, let $Y_{i},i\geq 1$ be ENOD random variables with common
distribution $G\in\mathcal{C}$, and let $Z_{i},\ i\geq 1$ be WOD random
variables with dominating coefficients $g_{U,H}(n)$ and $g_{L,H}(n),\ n\geq 1$
such that

$\displaystyle\lim_{t\to\infty}\max\\{g_{U,H}(n),g_{L,H}(n)\\}n^{-a}=0\ \ \ \
\ \ \text{for some}\ \ a\in(0,\infty).$ (2.4)

If $q_{1}=q_{2}=1$, $EZ_{1}^{r}\in(0,\infty)$ for some $r\in(1,\infty)$ and
$\overline{H}(x)=o\big{(}\overline{G}(x)\big{)}$, then (2.3) still holds for
each pair $\mu_{G}<\gamma<\Gamma<\infty$.

Without losing generality, we can set $a\in(1,\infty)$ in Theorem 2.3.

###  2.4 Precise large deviations of excess-of-net-loss process

Based on Theorem 2.1 and Theorem 2.2, we can study the precise large deviation
of some other objects of reinsurance, for example, excess-of-loss process

$R_{21}(t)=\sum\limits_{i=1}^{N(t)}(Y_{i}-D)^{+},\ \ \ \ t\in[0,\infty),$

where $D$ is some positive finite constant, see Example 5.2 $(ii)$ of
Klüppelberg and Mikosch [23]. Additionally, we believe that introducing income
factors will be better to evaluating reinsurance risks and determining more
reasonable limit $D$. Then, we focus on the corresponding research objects
excess-of-net-loss process

$R_{22}(t)=\sum_{i=1}^{N(t)}\big{(}(Y_{i}-D)^{+}-q_{2}cZ_{i}\big{)},\ \ \ \
t\in[0,\infty).$

In the following, we give three conclusions for $R_{22}(t)$. To this end, let
$G_{1}$ be distribution of $(Y_{1}-D)^{+}$, thus

$\overline{G_{1}}(x)=\textbf{1}_{(-\infty,0)}(x)+\overline{G}(x+D)\textbf{1}_{[0,\infty)}(x),\
\ \ x\in(-\infty,\infty)$

and

$\mu_{G_{1}}=E(Y_{1}-D)^{+}=EY_{1}\textbf{1}_{\\{Y_{1}>D\\}}-DP(Y_{1}>D).$

######  Theorem 2.4.

$(i)$ Under conditions as Theorem 2.1 $(i)$, for each
$\gamma\in(\mu_{G_{1}},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\mu_{H}^{-1}t,\infty)}\big{|}P\big{(}R_{22}(t)>x\big{)}\big{(}\mu_{H}^{-1}t\overline{G}(x+D-\mu_{G_{1}}\mu_{H}^{-1}t+q_{2}ct)\big{)}^{-1}-1\big{|}=0.$
(2.5)

$(ii)$ Under conditions as Theorem 2.2, for each pair
$\mu_{G}<\gamma<\Gamma<\infty$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma t,\Gamma
t]}\big{|}P\big{(}R_{22}(t)>x\big{)}\big{(}\mu_{H}^{-1}t\overline{G}(x+D-\mu_{G_{1}}\mu_{H}^{-1}t+q_{2}ct)\big{)}^{-1}-1\big{|}=0.$
(2.6)

$(iii)$ Under conditions as Theorem 2.3, (2.6) still holds for each pair
$\mu_{G}<\gamma<\Gamma<\infty$.

##  3 Two lemmas

Earlier work of the precise large deviation for nonrandom sums of independent
and identically distributed random variables with common distribution $F$ can
be found in Heyde
$\cite[cite]{[\@@bibref{}{H1967a}{}{}]},\cite[cite]{[\@@bibref{}{H1967b}{}{}]},\cite[cite]{[\@@bibref{}{H1968}{}{}]}$
and Nagaev
$\cite[cite]{[\@@bibref{}{N1969a}{}{}]},\cite[cite]{[\@@bibref{}{N1969b}{}{}]}$.
With the development of the research of precise large deviation, the classic
case of the distribution $F\in\mathcal{R}_{\alpha}$ on $[0,\infty)$ is
attributed to Nagaev
$\cite[cite]{[\@@bibref{}{N1973}{}{}]},\cite[cite]{[\@@bibref{}{N1979}{}{}]}$;
the case of $F\in\mathcal{ERV}(\alpha,\beta)$ can be referred to Cline and
Hsing [9], Klüppelberg and Mikosch [23], Mikosch and Nagaev [32], Tang et al.
[40], among others; and the case of $F\in\mathcal{C}$ is shown in Ng et al.
[37] and Wang and Wang [46]. For the corresponding results of local
probability and density, see Doney [12], Denisov et al. [11], Lin [28], Yang
et al. [50], Cheng and Li [7], etc.

For dependent random variables, there are also many corresponding results.
Konstantinides and Mikosch [25] obtained some important results on the precise
large deviation of sums for a stochastic recurrence equation; later, Mikosch
and Wintenberger [31] studied the same topic for a stationary regularly
varying sequence of random variables. Theorem 3.1 of Tang [39] studied the
precise large deviation of NOD random variables with common distribution
$F\in\mathcal{C}$ on $(-\infty,\infty)$. Thereafter, Theorem 2.1 of Liu [29]
replaced the condition $xF(-x)=o\big{(}\overline{F}(x)\big{)}$ in Tang [39]
with the condition $F(-x)=o\big{(}\overline{F}(x)\big{)}$ and gave the same
result for ENOD random variables.

Klüppelberg and Mikosch [23] and Chen et al. [5] gave some corresponding
results for random sums of independent and ENOD random variables,
respectively. In addition, Theorem 2.1 of Wang et al. [47] studied the precise
large deviation for $F\in\mathcal{D}$ on $[0,\infty)$.

Almost all of the above results require that the mean value of random
variables be zero. However, in order to prove Theorem 2.1-Theorem 2.2 of this
paper, we need to transform some of the above results into a noncentralized
form.

The following result is a noncentralized version for nonnegative random
variables of Theorem 2.2 of Liu [29].

######  Lemma 3.1.

Let $Y_{i},i\geq 1$ be ENOD random variables with common distribution
$G\in\mathcal{C}$ on $[0,\infty)$, dominating coefficient $M$ and mean
$\mu_{G}\in(0,\infty)$. Then for each $\gamma\in(\mu_{G},\infty)$,

$\displaystyle\lim\limits_{n\to\infty}\sup\limits_{x\in[\gamma
n,\infty)}\big{|}P(S^{G}_{n}>x)\big{(}n\overline{G}(x-\mu_{G}n)\big{)}^{-1}-1\big{|}=0.$
(3.1)

###### Proof.

Let $X_{1}=Y_{1}-\mu_{G}$ with distribution $F$. Clearly, $EX_{1}=0$,
$P(X_{1}^{-}>x)=o\big{(}\overline{F}(x)\big{)}$, $E(X_{1}^{-})^{r}<\infty$ for
some $r>1$ and

$P(S^{G}_{n}>x)=P\Big{(}\sum_{i=1}^{n}(Y_{i}-\mu_{G})>x-n\mu_{G}\Big{)}=P\Big{(}\sum_{i=1}^{n}X_{i}>x-n\mu_{G}\Big{)}.$

Further, for each $\gamma_{0}\in(0,\infty)$, let $\gamma=\gamma_{0}+\mu_{G}$.
Then $\gamma_{0}\in(0,\infty)$ is equivalent to $\gamma\in(\mu_{G},\infty)$.
Therefore, according to Theorem 2.2 of Liu [29], we have

$\displaystyle\lim\limits_{n\to\infty}\sup\limits_{x\in[\gamma
n,\infty)}\Big{|}\frac{P(S^{G}_{n}>x)}{n\overline{G}(x-\mu_{G}n)}-1\Big{|}=\lim\limits_{n\to\infty}\sup\limits_{x-n\mu_{G}\in[\gamma_{0}n,\infty)}\Big{|}\frac{P(S^{G}_{n}-n\mu_{G}>x-n\mu_{G})}{n\overline{G}(x-\mu_{G}n)}-1\Big{|}=0,$

that is (3.1) holds. $\Box$

######  Remark 3.1.

The restrict $\gamma\in(\mu_{G},\infty)$ has no effect on some applications of
the lemma, see, for example, on asymptotic behavior of random time ruin
probability in Theorem 5.2.

Then the following result comes from Theorem 3.1 of Klüppelberg and Mikosch
[23] for independent random variables and Theorem 3.1 of Chen et al. [5] for
ENOD random variables.

######  Lemma 3.2.

Let $Y_{i},i\geq 1$ be ENOD random variables with common distribution
$G\in\mathcal{C}$ on $[0,\infty)$, dominating coefficient $M$ and mean
$\mu_{G}\in(0,\infty)$, and let $M(t)$ be a nonnegative integer valued random
random with mean $m(t)=EM(t)\in(0,\infty)$ for all $t\in[0,\infty)$ such that
$\lim_{t\to\infty}m(t)=\infty$ and independent of $Y_{i},i\geq 1$. If for some
$p\in(J_{G}^{+},\infty)$ and each $\delta\in(0,\infty)$,

$\displaystyle\lim_{t\to\infty}EM^{p}(t)\emph{{1}}_{\\{M(t)\in((1+\delta)m(t),\infty)\\}}(t)m^{-1}(t)=0,$
(3.2)

then for each $\gamma\in(\mu_{G},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma
m(t),\infty)}\big{|}P(S^{G}_{M(t)}>x)\big{(}m(t)\overline{G}\big{(}x-\mu_{G}m(t)\big{)}\big{)}^{-1}-1\big{|}=0.$
(3.3)

######  Remark 3.2.

$(i)$ Many counting processes satisfy condition (3.2), such as, the counting
process in Lemma 2.1-Lemma 2.4 of Klüppelberg and Mikosch [23], and the quasi
renewal counting process in Proposition 1.4 of this paper.

$(ii)$ Clearly, condition (3.2) of this paper is equivalent to the condition
(3.2) in Theorem 3.1 of Chen et al. [5] that, for some
$p\in(J_{G}^{+},\infty)$ and each $\delta\in(0,\infty)$,

$\limsup_{t\to\infty}EM^{p}(t)\emph{{1}}_{\\{M(t)\in((1+\delta)m(t),\infty)\\}}m^{-1}(t)<\infty.$

######  Corollary 3.1.

In Case $i)$, let $Y_{i},i\geq 1$ be ENOD random variables with common
distribution $G\in\mathcal{C}$ and dominating coefficient $M$, and let
$Z_{i},\ i\geq 1$ be WLOD random variables with dominating coefficients
$g_{L,H}(n),\ n\geq 1$ satisfying condition (1.5). Then for each
$\gamma\in(\mu_{G},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\mu^{-1}_{H}t,\infty)}\big{|}P(S^{G}_{N(t)}>x)\big{(}\mu^{-1}_{H}t\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)\big{)}^{-1}-1\big{|}=0.$
(3.4)

###### Proof.

In Lemma 3.2, we replace $M(t)$ and $m(t)$ with $N(t)$ and $\lambda(t)$,
respectively. According to Proposition 1.5 and Proposition 1.6, by (1.5), we
know that $\lambda(t)\sim\mu_{H}^{-1}t$ as $t\to\infty$ and (1.4) holds. Thus,
according to Lemma 3.2, for each $\gamma\in(\mu_{G},\infty)$, we have

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\lambda(t),\infty)}\big{|}P(S^{G}_{N(t)}>x)\lambda^{-1}(t)\overline{G}^{-1}\big{(}x-\mu_{G}\lambda(t)\big{)}-1\big{|}=0.$
(3.5)

To prove (3.4), we first prove

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\lambda(t),\infty)}\big{|}\overline{G}\big{(}x-\mu_{G}\lambda(t)\big{)}\overline{G}^{-1}(x-\mu_{G}\mu^{-1}_{H}t)-1\big{|}=0.$
(3.6)

On the one hand, by $\lambda(t)\sim\mu_{H}^{-1}t$, $\gamma\in(\mu_{G},\infty)$
and $G\in\mathcal{C}$, for any $\varepsilon\in(0,1)$, when $t$ is large
enough, it holds uniformly for $x\in[\gamma\lambda(t),\infty)$ that

$\displaystyle\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)\overline{G}^{-1}\big{(}x-\mu_{G}\lambda(t)\big{)}\leq\overline{G}\big{(}x-\mu_{G}\lambda(t)(1+\varepsilon)\big{)}\overline{G}^{-1}\big{(}x-\mu_{G}\lambda(t)\big{)}$
$\displaystyle\leq$
$\displaystyle\overline{G}\big{(}(x-\mu_{G}\lambda(t))(1-\varepsilon\mu_{G}\lambda(t)(x-\mu_{G}\lambda(t))^{-1})\big{)}\overline{G}^{-1}\big{(}x-\mu_{G}\lambda(t)\big{)}$
$\displaystyle\leq$
$\displaystyle\overline{G}\big{(}(x-\mu_{G}\lambda(t))(1-\varepsilon\mu_{G}(\gamma-\mu_{G})^{-1})\big{)}\overline{G}^{-1}\big{(}x-\mu_{G}\lambda(t)\big{)}$
$\displaystyle\lesssim$
$\displaystyle\overline{G}_{*}^{-1}\big{(}(1-\varepsilon\mu_{G}(\gamma-\mu_{G})^{-1})^{-1}\big{)}$
$\displaystyle\downarrow$ $\displaystyle\overline{G}_{*}^{-1}(1)=1\ \ \ \ \ \
\ \text{as}\ \ t\to\infty,\ \ \text{thus}\ \ \varepsilon\downarrow 0.$

On the other hand, we can similarly prove that

$\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)\gtrsim\overline{G}\big{(}x-\mu_{G}\lambda(t)\big{)}$

uniformly for $x\in[\gamma\lambda(t),\infty)$ as $t\to\infty$. Thus, (3.6)
holds.

Combining (3.5) and (3.6), we know that

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\lambda(t),\infty)}\big{|}P(S^{G}_{N(t)}>x)\big{(}\mu^{-1}_{H}t\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)\big{)}^{-1}-1\big{|}=0.$
(3.7)

Now, we prove that (3.4) is equivalent to (3.7). On the one hand, if (3.4)
holds for each $\gamma\in(\mu_{G},\infty)$, then for each
$\gamma\in(\mu_{G},\infty)$, we take any
$\varepsilon\in(0,1-\gamma^{-1}\mu_{G})$ such that
$(1-\varepsilon)\gamma\in(\mu_{G},\infty)$ and

$\displaystyle\lim_{t\to\infty}\sup\limits_{x\in[\gamma\lambda(t),\infty)}\Big{|}\frac{P(S^{G}_{N(t)}>x)}{\mu^{-1}_{H}t\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)}-1\Big{|}\leq\lim_{t\to\infty}\sup\limits_{x\in[\gamma(1-\varepsilon)\mu^{-1}_{H}t,\infty)}\Big{|}\frac{P(S^{G}_{N(t)}>x)}{\mu^{-1}_{H}t\overline{G}(x-\mu_{G}\mu^{-1}_{H}t)}-1\Big{|}=0,$

that is (3.4)$\Rightarrow$(3.7). On the other hand, we can prove that
(3.7)$\Rightarrow$(3.4), similarly.

Therefore, (3.4) holds for each $\gamma\in(\mu_{G},\infty)$. $\Box$

##  4 Proofs of Theorem 2.1-Theorem 2.4

###  4.1 Proof of Theorem 2.1

$(i)$ On the one hand, we give the asymptotic upper bound of
$P\big{(}R_{12}(t)>x\big{)}$ for each $t\in(0,\infty)$.

For each $\gamma\in(q_{1}\mu_{G},\infty),\ t\in(0,\infty)$ and any
$\delta\in\big{(}0,\min\\{1,\mu_{H},\gamma
q_{1}^{-1}\mu_{G}^{-1}-1\\}\big{)}$, we split

$\displaystyle
P\big{(}R_{12}(t)>x\big{)}=\Big{(}\sum_{n<(1-\delta)t\mu_{H}^{-1}}+\sum_{(1-\delta)t\mu_{H}^{-1}\leq
n\leq(1+\delta)t\mu_{H}^{-1}}+\sum_{n>(1+\delta)t\mu_{H}^{-1})}\Big{)}P\big{(}S_{n}^{F}>q_{1}^{-1}x,N(t)=n\big{)}$
(4.1) $\displaystyle=$
$\displaystyle\Sigma_{1}(x,t)+\Sigma_{2}(x,t)+\Sigma_{3}(x,t).$

Before we deal with $\Sigma_{j}(x,t),\ 1\leq j\leq 3$, we need to give the
weak equivalence relations of several tail probabilities. When
$x\in[\gamma\mu_{H}^{-1}t,\infty)$, by $\gamma\in(q_{1}\mu_{G},\infty)$ and
the safety load condition $q_{1}\mu_{G}<q_{2}c\mu_{H}$, we know that

$\overline{G}(q_{1}^{-1}x)\leq\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu^{-1}_{H}t\big{)}\leq\overline{G}\big{(}q_{1}^{-1}x(1-q_{1}\mu_{G}\gamma^{-1})\big{)}$

and

$\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}\geq\overline{G}\big{(}q_{1}^{-1}x(1+(q_{2}c\mu_{H}-q_{1}\mu_{G})\gamma^{-1})\big{)}.$

The above two inequalities and $G\in\mathcal{C}\subset\mathcal{D}$ implies
that

$\displaystyle\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}\asymp\overline{G}(q_{1}^{-1}x)\asymp\overline{G}(q_{1}^{-1}x-\mu_{G}\mu^{-1}_{H}t),\
\ \ \ \ \text{as}\ t\to\infty.$ (4.2)

For $\Sigma_{1}(x,t)$, according to Corollary 3.1 and Proposition 1.6 (thus
Proposition 1.5), by (4.2), it holds uniformly for all
$x\in(\gamma\mu_{H}^{-1}t,\infty)$ that

$\displaystyle\Sigma_{1}(x,t)\leq
P\big{(}S_{\lfloor(1-\delta)\mu^{-1}_{H}t\rfloor}^{G}>q_{1}^{-1}x\big{)}P\big{(}N(t)\leq(1-\delta)\mu^{-1}_{H}t\big{)}$
(4.3) $\displaystyle\lesssim$
$\displaystyle(1-\delta)\mu^{-1}_{H}t\overline{G}\big{(}q_{1}^{-1}x-(1-\delta)\mu_{G}\mu^{-1}_{H}t\big{)}P\big{(}N(t)\leq(1-\delta)\mu^{-1}_{H}t\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu^{-1}_{H}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu^{-1}_{H}t)\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu^{-1}_{H}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t)\big{)},\
\ \ \ \ \ \text{as}\ t\to\infty.$

For $\Sigma_{3}(x,t)$, we recall Lemma 2.1 of Wang et al. [47] in the case of
NA random variables, which can also easily be generalized to the case of WUOD
random variable.

######  Lemma 4.1.

Let $Y_{i},i\geq 1$ be WUOD nonnegative random variables with common
distribution $G$, mean $\mu_{G}\in(0,\infty)$ and dominating coefficients
$g_{U,G}(n),\ n\geq 1$. Then, for each $n\geq 1$ and any $v\in(0,\infty)$,

$\displaystyle P(S^{G}_{n}>x)\leq
n\overline{G}(v^{-1}x)+g_{U,G}(n)(e\mu_{G}nx^{-1})^{v},\ \ \ \ \
x\in(0,\infty).$ (4.4)

Further, according to Lemma 4.4, Proposition 1.6 (thus Proposition 1.4 and
Proposition 1.5), Lemma 3.1 and Hölder inequality, by (4.4) with
$g_{U,G}(n)=M\geq 1,\ n\geq 1,\ v\in(J_{G}^{+},\infty),\
J_{G}^{+}\in[1,\infty)$ and (4.2), it holds uniformly for all
$x\in(\gamma\mu_{H}^{-1}t,\infty)$ that

$\displaystyle\Sigma_{3}(x,t)\leq\sum_{n>(1+\delta)\mu^{-1}_{H}t}P(S_{n}^{G}>q_{1}^{-1}x)P\big{(}N(t)=n\big{)}$
(4.5) $\displaystyle\leq$
$\displaystyle\sum_{n>(1+\delta)\mu^{-1}_{H}t}\big{(}n\overline{G}(v^{-1}q_{1}^{-1}x)+M(e\mu_{G}nx^{-1})^{v}\big{)}P\big{(}N(t)=n\big{)}$
$\displaystyle=$
$\displaystyle\overline{G}(v^{-1}q_{1}^{-1}x)EN(t)\textbf{1}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}+M(e\mu_{G}x^{-1})^{v}EN(t)N^{v-1}(t)\textbf{1}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}$
$\displaystyle=$ $\displaystyle
O\Big{(}\overline{G}(q_{1}^{-1}x)E^{2^{-1}}N^{2}(t)\big{(}P^{2^{-1}}(N(t)>(1+\delta)\mu_{H}^{-1}t)+E^{2^{-1}}N^{2(v-1)}(t)\textbf{1}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t)\big{)},\
\ \ \ \ \text{as}\ t\to\infty.$

For $\Sigma_{2}(x,t)$, according to Lemma 3.1, for any $0<\varepsilon<1$,
there exists $t_{0}=t_{0}(\varepsilon,G,q_{1},q_{2},c,\delta)>0$, when $t\geq
t_{0}$, it holds uniformly for all $x>\gamma\mu_{H}^{-1}t$ that

$\displaystyle\Sigma_{2}(x,t)=\sum_{\frac{(1-\delta)t}{\mu_{H}}\leq
n\leq\frac{(1+\delta)t}{\mu_{H}}}\int\cdots\int_{A_{n+1}(t)}P\Big{(}S_{n}^{G}>\frac{x}{q_{1}}+\frac{q_{2}c}{q_{1}}\sum_{i=1}^{n}z_{i}\Big{)}P(Z_{i}\in
dz_{i},1\leq i\leq n+1)$ (4.6) $\displaystyle\leq$
$\displaystyle(1+\varepsilon)\sum_{\frac{(1-\delta)t}{\mu_{H}}\leq
n\leq\frac{(1+\delta)t}{\mu_{H}}}\int\cdots\int_{A_{n+1}(t)}n\overline{G}\Big{(}\frac{x}{q_{1}}+\frac{q_{2}c}{q_{1}}\sum_{i=1}^{n}z_{i}-n\mu_{G}\Big{)}P(Z_{i}\in
dz_{i},1\leq i\leq n+1)$ $\displaystyle=$
$\displaystyle(1+\varepsilon)\Sigma_{0}(x,t),$

where $A_{n+1}(t)=A_{n+1}(t,H)=\\{\sum_{i=1}^{n}Z_{i}\leq
t,\sum_{i=1}^{n+1}Z_{i}>t\\},\ n\geq 1$.

To estimate $\Sigma_{0}(x,t)$, for the above $t$ and $\delta$, we denote

$B(t)=B(t,\delta,H)=\Big{\\{}(1-\delta)\mu_{H}^{-1}t\leq
N(t)\leq(1+\delta)\mu_{H}^{-1}t\Big{\\}}$

and

$D(t)=D(t,\delta,H)=\Big{\\{}(\mu_{H}-\delta)\lfloor(1-\delta)\mu_{H}^{-1}t\rfloor\leq\sum_{i=1}^{\lfloor(1-\delta)\mu_{H}^{-1}t\rfloor}Z_{i}\leq(\mu_{H}+\delta)\lfloor(1-\delta)\mu_{H}^{-1}t\rfloor\Big{\\}}.$

According to Lemma 2.1 of Chen et al. [5] and Proposition 1.6 (thus
Proposition 1.4), by (1.7) (thus (1.5)), we know that

$\displaystyle\lim_{t\to\infty}P\big{(}B(t)\big{)}=1\ \ \ \ \text{and}\ \ \ \
\ \lim_{t\to\infty}P\big{(}D(t)\big{)}=1.$ (4.7)

Then we again split

$\displaystyle\Sigma_{0}(x,t)\leq\sum_{(1-\delta)\mu_{H}^{-1}t\leq
n\leq(1+\delta)\mu_{H}^{-1}t}\Big{(}\int\cdots\int_{A_{n+1}(t)\cap
B^{c}(t)}+\int\cdots\int_{A_{n+1}(t)\cap B(t)}\Big{)}$ (4.8) $\displaystyle\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
n\overline{G}\Big{(}xq_{1}^{-1}+q_{2}cq_{1}^{-1}\sum_{i=1}^{n}z_{i}-(1+\delta)\mu_{H}^{-1}\mu_{G}t\Big{)}P(Z_{i}\in
dz_{i},1\leq i\leq n+1)$ $\displaystyle=$
$\displaystyle\Sigma_{01}(x,t)+\Sigma_{02}(x,t).$

We first deal with $\Sigma_{01}(x,t)$. For the above $\gamma$ and $\delta$, we
have $\gamma\in(q_{1}(1+\delta)\mu_{G},\infty)$. Then according to Hölder
inequality and Proposition 1.6 (thus Proposition 1.5), by (4.7),
$G\in\mathcal{C}\subset\mathcal{D}$ and (4.2), it holds uniformly for all
$x\in(\gamma\mu_{H}^{-1}t,\infty)$ that

$\displaystyle\Sigma_{01}(x,t)\leq\overline{G}\big{(}q_{1}^{-1}x-(1+\delta)\mu_{G}t\mu_{H}^{-1}\big{)}EN(t)\textbf{1}_{B^{c}(t)}$
(4.9) $\displaystyle\leq$
$\displaystyle\overline{G}\big{(}q_{1}^{-1}x-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}E^{2^{-1}}N^{2}(t)P^{2^{-1}}\big{(}B^{c}(t)\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x(1-(1+\delta)\mu_{G}\mu_{H}^{-1}tx^{-1}q_{1}))\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x(1-(1+\delta)q_{1}\mu_{G}\gamma^{-1}))\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x)\big{)}$ $\displaystyle=$
$\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t)\big{)},\
\ \ \ \ \ \text{as}\ t\to\infty.$

Next, we deal with $\Sigma_{02}(x,t)$. For the above $t$ and $\delta$, we
continue to separate

$\displaystyle\Sigma_{02}(x,t)=\sum_{(1-\delta)t\mu_{H}^{-1}\leq
n\leq(1+\delta)t\mu_{H}^{-1}}\Big{(}\int\cdots\int_{A_{n+1}\cap B(t)\cap
D^{c}(t)}+\int\cdots\int_{A_{n+1}\cap B(t)\cap D(t)}\Big{)}$ (4.10)
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
n\overline{G}\Big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}\sum_{i=1}^{n}z_{i}-n\mu_{G}\Big{)}P(Z_{i}\in
dz_{i},1\leq i\leq n+1)$ $\displaystyle=$
$\displaystyle\Sigma_{021}(x,t)+\Sigma_{022}(x,t).$

According to Hölder inequality and Proposition 1.6 (thus Proposition 1.5)
again, by (4.7), $\gamma\in(q_{1}(1+\delta)\mu_{G},\infty)$, (4.2) and
$G\in\mathcal{C}\subset\mathcal{D}$, it holds uniformly for all
$x\in[\gamma\mu^{-1}_{H}t,\infty)$ that

$\displaystyle\Sigma_{021}(x,t)\leq\overline{G}\big{(}q_{1}^{-1}x-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}EN(t)\textbf{1}_{D^{c}(t,\delta)}$
(4.11) $\displaystyle\leq$
$\displaystyle\overline{G}\big{(}q_{1}^{-1}x-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}E^{2^{-1}}N^{2}(t)P^{2^{-1}}\big{(}D^{c}(t)\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x(1-(1+\delta)\mu_{G}\mu_{H}^{-1}tx^{-1}q_{1}))\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x(1-(1+\delta)\mu_{G}\gamma^{-1}q_{1}))\big{)}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x)\big{)}$ $\displaystyle=$
$\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t)\big{)},\
\ \ \ \ \ \ \text{as}\ t\to\infty.$

For $c,\ t_{0},\ \varepsilon,\ \delta$ and each
$\gamma\in(q_{1}(1+\delta)\mu_{G},\infty)$ mentioned above, by the definitions
of $B(t)$ and $D(t)$, there exists
$t_{1}=t_{1}(\varepsilon,G,q_{1},q_{2},c,\delta)\in[t_{0},\infty)$ such that,
when $t\in[t_{1},\infty)$, it holds uniformly for all
$x\in[\gamma\mu^{-1}_{H}t,\infty)$ that

$\displaystyle\Sigma_{022}(x,t)\leq\sum_{\frac{(1-\delta)t}{\mu_{H}}\leq
n\leq\frac{(1+\delta)t}{\mu_{H}}}\int\cdots\int_{A_{n+1}\cap B(t)\cap
D(t)}n\overline{G}\Big{(}\frac{x}{q_{1}}+\frac{q_{2}c}{q_{1}}\sum_{i=1}^{\lfloor\frac{(1-\delta)t}{\mu_{H}}\rfloor}z_{i}-\frac{(1+\delta)\mu_{G}t}{\mu_{H}}\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
P(Z_{i}\in dz_{i},1\leq i\leq n+1)$ $\displaystyle\leq$
$\displaystyle\overline{G}\big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}(1-\delta)\mu_{H}((1-\delta)\mu_{H}^{-1}t-1)-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}EN(t)$
$\displaystyle<$
$\displaystyle(1+\varepsilon)\mu_{H}^{-1}t\overline{G}\big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}(1-\delta)^{2}t-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}.$

Here, by the safety load condition $q_{1}\mu_{G}<q_{2}c\mu_{H}$, we know that
for above $\varepsilon$ and $\delta$, we have

$0<q_{2}cq_{1}^{-1}(1-\delta)^{2}-(1+\delta)\mu_{G}\mu_{H}^{-1}\uparrow
q_{2}cq_{1}^{-1}-\mu_{G}\mu_{H}^{-1},\ \ \ \ \ \ \text{as}\ \delta\downarrow
0.$

Further, let $U$ be a uniform distribution on $[0,1]$, then
$\overline{G*U}(\cdot)$ is a uniformly continuous function on
$(-\infty,\infty)$ and

$\overline{G*U}(x)\sim\overline{G}(x).$

Therefore, for the above $\varepsilon$, there exists
$\delta_{0}\in(0,\min\\{1,\mu_{H},\gamma\mu_{G}^{-1}-1\\})$ and
$t_{2}=t_{2}(\varepsilon,G,q_{1},q_{2},c,\delta)\in[t_{1},\infty)$ such that,
when $\delta\in(0,\delta_{0})$ and $t\in[t_{2},\infty)$, by the above three
facts, it holds uniformly for all $x\in[\gamma\mu^{-1}_{H}t,\infty)$ that

$\displaystyle\Sigma_{022}(x,t)\leq(1+\varepsilon)^{2}\mu_{H}^{-1}t\overline{G*U}\big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}(1-\delta)^{2}t-(1+\delta)\mu_{G}\mu_{H}^{-1}t\big{)}$
(4.12) $\displaystyle<$
$\displaystyle(1+\varepsilon)^{3}\mu_{H}^{-1}t\overline{G*U}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}$
$\displaystyle<$
$\displaystyle(1+\varepsilon)^{4}\mu_{H}^{-1}t\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}.$

Combined with (4.1), (4.3)-(4.6) (4.8)-(4.12), and by arbitrariness of
$\varepsilon$, it holds uniformly for all $x\in[\gamma\mu^{-1}_{H}t,\infty)$
that

$\displaystyle
P\big{(}R_{12}(t)>x\big{)}\lesssim\mu_{H}^{-1}t\overline{G}\big{(}xq_{1}^{-1}-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)},\
\ \ \ \ \ \ as\ t\to\infty.$ (4.13)

On the other hand, we continue to find the asymptotic lower bound of
$P\big{(}R_{12}(t)>x\big{)}$. From (4.9) and (4.11), we know that

$EN(t)\textbf{1}_{B^{c}(t)\cup
D^{c}(t)}=o\big{(}\mu_{H}^{-1}t\big{)}=o\big{(}EN(t)\big{)},\ \ \ \text{as}\
t\to\infty.$

Therefore, for the above $\delta$ and $\varepsilon$, there exists
$t_{3}=t_{3}(\varepsilon,G,q_{1},q_{2},c,\delta)\in[t_{2},\infty)$ such that,
when $t\in[t_{3},\infty)$, for each
$\gamma\in(q_{1}(1+\delta)\mu_{G},\infty)$, it holds uniformly for all
$x\in(\gamma\mu_{H}^{-1}t,\infty)$ that

$\displaystyle P\big{(}R_{12}(t)>x\big{)}\geq\Sigma_{2}(x,t)$ (4.14)
$\displaystyle\geq$ $\displaystyle(1-\varepsilon)\Sigma_{0}(x,t)$
$\displaystyle\geq$ $\displaystyle(1-\varepsilon)\Sigma_{022}(x,t)$
$\displaystyle\geq$ $\displaystyle\sum_{(1-\delta)t\mu_{H}^{-1}\leq
n\leq(1+\delta)t\mu_{H}^{-1}}\int\cdots\int_{A_{n+1}\cap B(t)\cap
D(t)}n\overline{G}\Big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}\sum_{i=1}^{n}z_{i}-(1-\delta)\mu_{G}t\mu_{H}^{-1}\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
P(Z_{i}\in dz_{i},1\leq i\leq n+1)$ $\displaystyle\geq$
$\displaystyle\overline{G}\big{(}q_{1}^{-1}x+q_{2}cq_{1}^{-1}(1+\delta)\mu_{H}(1+\delta)t\mu_{H}^{-1}-(1-\delta)\mu_{G}\mu_{H}^{-1}t\big{)}\big{(}EN(t)-EN(t)\textbf{1}_{B^{c}(t)\cup
D^{c}(t)}\big{)}$ $\displaystyle>$
$\displaystyle(1-\varepsilon)\mu_{H}^{-1}t\overline{G}\big{(}q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t+q_{2}cq_{1}^{-1}t\big{)}.$

Combined with (4.1), (4.13), (4.14) and arbitrariness of $\varepsilon$, for
each $\gamma\in(q_{1}\mu_{G},\infty)$, (2.1) holds.

$(ii)$ According to Proposition 1.4, by (1.5) (thus (1.3)), we know that (1.4)
holds. Then, according to Corollary 3.1, for each
$\gamma\in(q_{1}\mu_{G},\infty)$, it holds uniformly for all
$x\in[\gamma\mu^{-1}_{H}t,\infty)$ that

$\displaystyle
P\big{(}R_{11}(t)>x\big{)}=P(S^{G}_{N(t)}>q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t)\sim\mu_{H}^{-1}t\overline{G}(q_{1}^{-1}x-\mu_{G}\mu_{H}^{-1}t),\
\ \ \ \ \ \ \text{as}\ t\to\infty,$

that is (2.2) holds.

###  4.2 Proof of Theorem 2.2

Recall $B(t)$ and $D(t)$ mentioned above. Similarly, let

$D^{*}(t)=\big{\\{}(1-\delta)\mu_{H}\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor\leq\sum_{i=1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}Z_{i}\leq(1+\delta)\mu_{H}\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor\big{\\}},\
\ \ \ t\in[0,\infty).$

For large enough $t\in[0,\infty)$, on the one hand, we have

$\displaystyle P\big{(}R_{12}(t)>x\big{)}\geq
P\Big{(}\sum_{i=1}^{N(t)}Y_{i}>x+c\sum_{i=1}^{N(t)}Z_{i},B(t)\cap
D^{*}(t)\Big{)}$ (4.15) $\displaystyle\geq$ $\displaystyle
P\Big{(}\sum_{i=1}^{(1-\delta)\mu_{H}^{-1}t}Y_{i}-\mu_{G}\mu_{H}^{-1}t>x+c(1+\delta)^{2}t-\mu_{G}\mu_{H}^{-1}t,B(t)\cap
D^{*}(t)\Big{)}$ $\displaystyle\geq$ $\displaystyle
P\Big{(}\sum_{i=1}^{(1-\delta)\mu_{H}^{-1}t}Y_{i}>x+c(1+\delta)^{2}t\Big{)}-P\big{(}B^{c}(t)\big{)}-P\big{(}D^{*c}(t)\big{)}.$

On the other hand, we have

$\displaystyle P\big{(}R_{12}(t)>x\big{)}\leq
P\Big{(}\sum_{i=1}^{N(t)}Y_{i}>x+c\sum_{i=1}^{N(t)}Z_{i},B(t)\cap
D^{*}(t)\Big{)}+P\big{(}B^{c}(t)\big{)}+P\big{(}D^{*c}(t)\big{)}$ (4.16)
$\displaystyle\leq$ $\displaystyle
P\Big{(}\sum_{i=1}^{(1+\delta)\mu_{H}^{-1}t}Y_{i}>x+c(1-\delta)^{2}t\Big{)}+P\big{(}B^{c}(t)\big{)}+P\big{(}D^{*c}(t)\big{)}.$

Further, when $x\in[\gamma t,\Gamma t]$, by (4.15) and (4.16) in this paper
and (4.7) in Chen et al. [6], we have

$\displaystyle\max\\{P\big{(}B^{c}(t)\big{)},P\big{(}D^{*c}(t)\big{)}\\}=o\big{(}\mu_{H}^{-1}t\overline{G}(x)\big{)}=o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct)\big{)},\
\ \ \text{as}\ t\to\infty.$ (4.17)

Therefore, according to Lemma 3.1, by (4.15), (4.16) and (4.17), we have

$\displaystyle P\big{(}R_{12}(t)>x\big{)}\gtrsim
P\Big{(}\sum_{i=1}^{(1-\delta)\mu_{H}^{-1}t}Y_{i}>x+c(1+\delta)^{2}t\Big{)}$
(4.18) $\displaystyle\sim$
$\displaystyle(1-\delta)\mu_{H}^{-1}t\overline{G}\big{(}x-(1-\delta)\mu_{G}\mu_{H}^{-1}t+c(1+\delta)^{2}t\big{)}$

and

$\displaystyle P\big{(}R_{12}(t)>x\big{)}\lesssim
P\Big{(}\sum_{i=1}^{(1+\delta)\mu_{H}^{-1}t}Y_{i}>x+c(1-\delta)^{2}t\Big{)}$
(4.19) $\displaystyle\sim$
$\displaystyle(1+\delta)\mu_{H}^{-1}t\overline{G}\big{(}x-(1+\delta)\mu_{G}\mu_{H}^{-1}t+c(1-\delta)^{2}t\big{)},\
\ \ \text{as}\ t\to\infty.$

More exactly, using the proof method of (4.12), combining with (4.18) and
(4.19), we obtain (2.3).

###  4.3 Proof of Theorem 2.3

From the proof of Theorem 2.2, we already know that (4.15) and (4.16) hold.
Therefore, we only need to prove (4.17). And by the proof of Theorem 1.1 of
Chen et al. [6], we just need to deal with $D^{*c}(t)$ in (4.17). To this end,
we first introduce some corresponding concepts and their properties.

Let $g(\cdot)$ be a positive function on $[0,\infty)$. Further, we define some
functions on $[0,\infty)$ as follows:

$\displaystyle
f_{g}(x)=\frac{1}{g(x)}\Big{(}\int_{0-}^{x}+\int_{(-x)-}^{0}\Big{)}|y|g(y)F(dy)=f_{g}^{+}(x)+f_{g}^{-}(x)$

and

$\displaystyle
F_{g}(x)=\big{(}f_{g}^{+}(x)+xP(X_{1}>x)\big{)}+\big{(}f_{g}^{-}(x)+xP(X_{1}<-x)\big{)}=F_{g}^{+}(x)+F_{g}^{-}(x).$

Then, we characterize the convergence rate of the function $F_{g}^{+}(\cdot)$
and the upper bound of $P(S_{n}>x)$. See Proposition 1 and (2.10) in Theorem 1
of Chen et al. [4], respectively.

######  Lemma 4.2.

For $j=1,2$, let $g_{j}(\cdot)$ on $[0,\infty)$ be a positive even function
such that $g_{j}(x)\underline{\uparrow}\infty$, and let function
$g(\cdot)=g_{1}(\cdot)g_{2}(\cdot)$. If $EX_{1}^{+}g_{1}(X_{1})\in(0,\infty)$,
then

$\displaystyle\lim g_{1}(x)F_{g}^{+}(x)=0.$ (4.20)

######  Lemma 4.3.

Let $X_{i},i\geq 1$ be WUOD random variables with common distribution $F$ on
$(-\infty,\infty)$, mean $EX_{1}=0$ and with dominating coefficients
$g_{U}(n),g_{L}(n),\ n\geq 1$, $g_{1}(\cdot)$ and $g_{2}(\cdot)$ be two
positive even functions on $[0,\infty)$ such that
$g_{j}(x)\underline{\uparrow}\infty,\ j=1,2$, and
$g(\cdot)=g_{1}(\cdot)g_{2}(\cdot)$. If $EX_{1}^{+}g_{1}(X_{1})\in(0,\infty)$
and there is some positive integer $m$ such that

$\displaystyle x^{m}g^{-1}(x)\underline{\uparrow}\infty\ \ \ \ \ \text{and}\ \
\ \ \ \ x^{m-1}g_{1}^{-1}(x)\overline{\downarrow}0,$ (4.21)

then for each $\gamma\in(0,\infty)$ and each pair $v,\theta\in(0,1)$, there
are positive constants $x_{0}=x_{0}\big{(}F,m,v,\theta,$
$g_{1}(\cdot),g_{2}(\cdot)\big{)}$ and
$C_{0}=C_{0}\big{(}F,m,v,\theta,g_{1}(\cdot),g_{2}(\cdot)\big{)}$ such that
for all $n\geq 1$,

$\displaystyle P(S_{n}>x)\leq
n\overline{F}(vx)+C_{0}g_{U,F}(n)\big{(}F_{g}^{+}(vx)\big{)}^{\frac{1-\theta}{v}}\
\ \ \text{for all}\ \ x\in[\max\\{\gamma n,x_{0}\\},\infty).$ (4.22)

For simplicity, we set $C_{0}=1$. Now, we prove Theorem 2.3.

On the one hand, let $H_{0}$ be distribution of $Z_{1,0}=Z_{1}-\mu_{H}$.
Clearly, $Z_{i,0},i\geq 1$ still are WOD random variables with the same
dominating coefficient as $Z_{i},i\geq 1$. For $r\in(1,\infty)$, there exists
a positive integer $m$ such that $r\in(1,2),m=1$ or $r\in[m,m+1),m>1$. Because
$EZ_{1}^{r}\in(0,\infty)$, there is a positive even functions
$g_{2,H_{0}}(\cdot)$ on $[0,\infty)$ such that

$g_{2,H_{0}}(x)\underline{\uparrow}\infty,\ \ \ \
x^{m+1-r}g^{-1}_{2,H_{0}}(x)\underline{\uparrow}\infty\ \ \ \text{and}\ \ \ \
E|Z_{1,0}|^{r}g_{2,H_{0}}(Z_{1,0})\in(0,\infty).$

Then we take $g_{1,H_{0}}(x)=|x|^{r-1},\ x\in(-\infty,\infty)$ and
$g_{H_{0}}(\cdot)=g_{1,H_{0}}(\cdot)\cdot g_{2,H_{0}}(\cdot)$. Clearly,
$E|Z_{1,0}|g_{1,H_{0}}$ $(Z_{1,0})\in(0,\infty)$ and (4.21) holds for
$g_{H_{0}}(\cdot)$ and $g_{1,H_{0}}(\cdot)$. Further, according to Lemma 4.2
and Lemma 4.3, we respectively know that

$\displaystyle H_{0,g}^{+}(x)=o\big{(}g_{1,H_{0}}^{-1}(x)\big{)}=o(x^{1-r})$
(4.23)

and for each $\gamma\in(0,\infty)$, each pair $\theta,v\in(0,1)$, there is
some positive constant $x_{0}=x_{0}\big{(}H_{0},m,v,\theta,$ $g_{1}(\cdot),$
$g_{2}(\cdot)\big{)}$ such that for all $n\geq 1$ and $x\in[\max\\{\gamma
n,x_{0}\\},\infty)$,

$\displaystyle P(S^{H}_{n}-n\mu_{H}>x)=P(S^{H_{0}}_{n}>x)\leq
n\overline{H_{0}}(vx)+g_{U,H_{0}}(n)\big{(}H_{0,g}^{+}(vx)\big{)}^{(1-\theta)v^{-1}}.$
(4.24)

For each $p\in(J_{F}^{+},\infty)$, we take integer $k$ large enough and
$v=(1-\theta)^{k+1}$ such that

$\displaystyle(r-1)(1-\theta)^{-k}>a-1+p.$ (4.25)

Further, according to Proposition 1.2 $(iii)$, by $G\in\mathcal{C}$,
$g_{U,H_{0}}(n)=g_{U,H}(n)=o(n^{a})$ as $n\to\infty$, (4.23), (4.25) and
$r\in(1,\infty)$, we have

$\displaystyle
g_{U,H_{0}}(n)\big{(}H_{0,g}^{+}(vx)\big{)}^{(1-\theta)v^{-1}}\big{(}n\overline{G}(x)\big{)}^{-1}=o\big{(}n^{a-1}x^{p-(r-1)(1-\theta)^{-k}}\big{)}$
(4.26) $\displaystyle=$ $\displaystyle
o\big{(}x^{a-1+p-(r-1)(1-\theta)^{-k}}\big{)}\to 0,\ \ \ \ \ \ \ \text{as}\
n\to\infty.$

And by $G\in\mathcal{C}\subset\mathcal{D}$ and
$\overline{H}(x)=o\big{(}\overline{G}(x)\big{)}$, we know that for all
$x\geq\gamma n$,

$\displaystyle
n\overline{H_{0}}(vx))=o\big{(}n\overline{G}(vx)\big{)}=o\big{(}n\overline{G}(x)\big{)}.$
(4.27)

Combining (4.24), (4.26), (4.27) and (4.2), by $x\in[\gamma t,\Gamma t]$, for
any $\delta\in(0,\mu_{H})$, we have

$\displaystyle
P\Big{(}\sum_{i=1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}(Z_{i}-\mu_{H})>\delta\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor\Big{)}=o\big{(}\mu_{H}^{-1}t\overline{G}(\delta\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor)\big{)}$
(4.28) $\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct)\big{)}.$

On the other hand, because $Z_{i},i\geq 1$ are WOD random variables,
$\mu_{H}-Z_{i},i\geq 1$ still are WUOD random variables with common
distribution $H_{1}$ and the dominating coefficients
$g_{U,H_{1}}=g_{L,H},n\geq 1$. Along the line certified above, by
$\overline{H_{1}}(x)=0$ for $x\in[\mu_{H},\infty)$, we have

$\displaystyle
P\Big{(}\sum_{i=1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}(Z_{i}-\mu_{H})<-\delta\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor\Big{)}=P\Big{(}\sum_{i=1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}(\mu_{H}-Z_{i})>\delta\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor\Big{)}$
(4.29) $\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct)\big{)}.$

Combining (4.28) and (4.29), we know that

$P\big{(}D^{*c}(t)\big{)}=o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct)\big{)}\
\ \text{as}\ \ t\to\infty.$

###  4.4 Proof of Theorem 2.4

$(i)$ According to Theorem 2.1 and Theorem 2.2 with $q_{1}=1$ and
$q_{2}\in(0,1]$, by the facts

$\displaystyle P\big{(}(Y_{1}-D)^{+}>x\big{)}=\overline{G}(x+D)\ \ \ \
\text{for}\ x\in[0,\infty)$

and

$\displaystyle
P\big{(}R_{22}(t)>x\big{)}=P\bigg{(}\sum_{i=1}^{N(t)}\big{(}(Y_{i}-D)^{+}-q_{2}cZ_{i}\big{)}>x\bigg{)}\
\ \ \ \text{for}\ x\in[0,\infty),$

we immediately get (2.5) and (2.6), respectively.

The proofs of $(ii)$ and $(iii)$ are similar to that of $(i)$, so their
details are omitted.

##  5 Two applications

###  5.1 Asymptotic estimate of mean of stop-loss reinsurance process

We denote stop-loss reinsurance process

$R_{31}(t)=\bigg{(}\frac{\sum_{i=1}^{N(t)}Y_{i}}{c\mu_{G}\lambda(t)}-K\bigg{)}^{+},\
\ \ \ t\in[0,\infty),$

where $K\in(c^{-1},\infty)$ and $c$ is the premium income mentioned above, see
Example 5.2 $(iii)$ of Klüppelberg and Mikosch [23]. However, we believe that
the premium income $c$ has no obvious significance here. Therefore, we may as
well removes this factor and replace $\lambda(t)$ with $\mu_{H}^{-1}t$ in the
following. Further, for $q_{1}\in(0,1]$ and $q_{2}\in[0,1]$, by the safety
load condition, we have

$\mu_{0}=|q_{2}\mu_{H}-q_{1}c\mu_{G}|>0.$

Then we focus on the corresponding research objects stop-net-loss reinsurance
process

$\displaystyle
R_{32}(t)=\bigg{(}\frac{\sum_{i=1}^{N(t)}(q_{1}Y_{i}-q_{2}cZ_{i})}{\mu_{0}\mu_{H}^{-1}t}-K\bigg{)}^{+},\
\ \ \ \ t\in[0,\infty).$ (5.1)

To this end, we recall the integral tail distribution of distribution $G$

$G_{I}(x)=\mu_{G}^{-1}\int_{0-}^{x}\overline{G}(y)dy\textbf{1}_{[0,\infty)}(x),\
\ \ \ x\in(-\infty,\infty),$

see Klüppelberg [22].

######  Theorem 5.1.

Assume that $K\in(q_{1}\mu_{0}^{-1}\mu_{G},\infty)$ when $q_{2}=0$ or
$K\in(0,\infty)$ when $q_{2}\in(0,1]$.

$(i)$ Under conditions as Theorem 2.1 $(i)$, it holds that

$\displaystyle\lim\limits_{t\to\infty}\frac{ER_{32}(t)}{q_{1}\mu_{0}^{-1}\mu_{G}\overline{G_{I}}(q_{1}^{-1}\mu_{0}\mu_{H}^{-1}Kt-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct)}=1.$
(5.2)

$(ii)$ Under conditions as Theorem 2.2, (5.2) still holds.

$(iii)$ Under conditions as Theorem 2.3, (5.2) still holds.

######  Remark 5.1.

$(i)$ In particular, if $q_{1}=1$ and $q_{2}=0$, then $\mu_{0}=\mu_{G}$ and
$R_{32}(t)=R_{31}(t)$. Further, if $G\in\mathcal{R}_{\alpha}$ for some
$\alpha\in(1,\infty)$, then according to $(i)$ and $(ii)$ of Theorem 5.1 and
Karamata Theorem, we can obtain the corresponding results in Corollary 3.4 of
Klüppelberg and Mikosch [23] and Corollary 2.1 of Chen et al. [6],
respectively.

$(ii)$ Theorem 5.1 also contains some new results.

######  Example 5.1.

In Theorem 5.1, let

$\overline{G}(x)=\emph{{1}}_{(-\infty,0)}(x)+(1+x)^{-1}\ln^{-2}(e+x)\emph{{1}}_{[0,\infty)}(x),\
\ \ x\in(-\infty,\infty).$

Clearly, $G\in\mathcal{R}_{1}$, $\mu_{G}\in(0,\infty)$,
$EY_{1}^{\alpha}=\infty$ for each $\alpha\in(1,\infty)$ and

$\overline{G}(x)\sim x^{-1}\ln^{-2}x.$

Then according to Theorem 5.1, when $t\to\infty$, we have

$\displaystyle
ER_{32}(t)\sim\frac{q_{1}}{\mu_{0}}\int_{\frac{\mu_{0}K}{q_{1}\mu_{H}}-\frac{\mu_{G}t}{\mu_{H}}+\frac{q_{2}ct}{q_{1}}}^{\infty}\frac{1}{v\ln^{2}v}dv=\frac{q_{1}\mu_{0}^{-1}}{\ln\big{(}(\frac{\mu_{0}K}{q_{1}\mu_{H}}-\frac{\mu_{G}}{\mu_{H}}+\frac{q_{2}c}{q_{1}})t\big{)}}\sim\frac{q_{1}}{\mu_{0}}\overline{G}(t)t\ln
t\asymp ER_{31}(t).$

Here, we find an interesting phenomenon that $c,\ K$ and $H$ do not affect the
asymptotic expression of $ER_{31}(t)$ and $ER_{32}(t)$. For the case that
$\alpha\in(1,\infty)$, however, they all play a role in the above expression.
In addition, when $\alpha=1$, $ER_{31}(t),\ ER_{32}(t)$ and
$\overline{G}(t)t\ln t$ are infinitesimals of the same order, which is
different from the case $\alpha\in(1,\infty)$ with the same order as
$\overline{G}(t)t$, as $t\to\infty$.

###### Proof.

We only prove $(i)$. By the safety load condition that
$q_{1}\mu_{G}<q_{2}c\mu_{H}$ and $K\in(q_{1}\mu_{0}^{-1}\mu_{G},\infty)$ when
$q_{2}=0$,

$\lim_{t\to\infty}q_{1}^{-1}\mu_{0}\mu_{H}^{-1}Kt-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct=\infty.$

Then according to Theorem 2.1, we have

$\displaystyle
ER_{32}(t)=\int_{0-}^{\infty}P\bigg{(}\sum_{i=1}^{N(t)}(q_{1}Y_{i}-q_{2}cZ_{i})>\mu_{0}\mu_{H}^{-1}t(K+u)\bigg{)}du$
$\displaystyle\sim$
$\displaystyle\mu^{-1}_{H}t\int_{0-}^{\infty}\overline{G}\big{(}\mu_{0}\mu_{H}^{-1}tq_{1}^{-1}(K+u)-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct\big{)}du$
$\displaystyle=$
$\displaystyle(q_{1}^{-1}\mu_{0})^{-1}\int_{q_{1}^{-1}\mu_{0}\mu_{H}^{-1}Kt-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct}^{\infty}\overline{G}(v)dv$
$\displaystyle=$ $\displaystyle
q_{1}\mu_{0}^{-1}\mu_{G}\mu_{G}^{-1}\int_{q_{1}^{-1}\mu_{0}\mu_{H}^{-1}Kt-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct}^{\infty}\overline{G}(v)dv,\
\ \ \ \ \text{as}\ \ t\to\infty$ $\displaystyle=$ $\displaystyle
q_{1}\mu_{0}^{-1}\mu_{G}\overline{G_{I}}(q_{1}^{-1}\mu_{0}\mu_{H}^{-1}Kt-\mu_{G}\mu_{H}^{-1}t+q_{1}^{-1}q_{2}ct).$

###  5.2 Asymptotic estimate of random time ruin probability

In a renewal risk model, the finite-time ruin probability at some time
$t\in[0,\infty)$ is defined by

$\displaystyle\psi(x;t)=P\Big{(}R_{4}(t)>x\Big{)},\ \ \ \ \ x\in[0,\infty),$
(5.3)

where the maximum net loss process

$R_{4}(t)=\sup\limits_{0\leq n\leq N(t)}S^{F}_{n}=\sup\limits_{0\leq n\leq
N(t)}\sum_{i=1}^{n}X_{i}=\sup\limits_{0\leq n\leq
N(t)}\sum_{i=1}^{n}(Y_{i}-cZ_{i}),\ \ \ t\in[0,\infty).$

Therefore, the random time ruin probability, as a more reasonable research
object, is defined by

$\displaystyle\psi(x;\tau)=P\Big{(}R_{4}(\tau)=\sup\limits_{0\leq n\leq
N(\tau)}S^{F}_{n}>x\Big{)},\ \ \ \ \ x\in[0,\infty),$ (5.4)

where $\tau$ is a nonnegative random variable mentioned above. See Remark 5.2
below for relevant researches. Here, we give a new result on asymptotic
estimate of the random time ruin probability.

######  Theorem 5.2.

In Case $i)$, let $Y_{i},i\geq 1$ be ENOD random variables and $Z_{i},i\geq 1$
be WLOD random variables satisfying condition (1.3). If $G\in\mathcal{C}$ and

$\displaystyle P(\tau>x)=o\big{(}\overline{G}(x)\big{)},$ (5.5)

then $\max\\{E\tau,E\lambda(\tau)\big{(}=EN(\tau)\big{)}\\}<\infty$ and

$\displaystyle\psi(x;\tau)\sim E\lambda(\tau)\overline{G}(x).$ (5.6)

######  Remark 5.2.

We make a comparison between the above theorem and the existing related
results.

In Theorem 2.1 of Wang et al. [45] and Corollary 2.2 (2) of Wang et al. [44],
random variables $Y_{i},i\geq 1$ are independent and identically distributed.
In addition, the former requires $Ee^{s_{0}N(\tau)}\in(0,\infty)$ for some
$s_{0}\in(0,\infty)$, and the latter requires $EZ_{1}^{p}<\infty$ for some
$p\in[2,\infty)$ or other.

In Theorem 3.1, Theorem 3.2, Theorem 3.3 and Theorem 3.4 of Wang et al. [45],
random variables $Y_{i},i\geq 1$ are NUOD and $Z_{i},i\geq 1$ are NOD. In
addition, the above Theorem 3.1 and Theorem 3.2 require
$EY_{1}^{1+\beta}\in(0,\infty)$ for some $\beta\in(0,\infty)$; the above
Theorem 3.3 and Theorem 3.4 require $EN^{p}(\tau)\in(0,\infty)$ for some
$p\in(J_{G}^{+},\infty)$, where $J_{G}^{+}\in[1,\infty)$, which is stronger
than condition (5.5), see Proposition 5.1 below.

Clearly, Theorem 5.2 of this paper has more advantages than the existing
results mentioned above. This is because we mainly make use method of precise
large deviation to prove the theorem, which is different from that of the
above existing results.

###### Proof.

Clearly, by (5.5) and $\mu_{G}\in(0,\infty)$, we know that
$E\tau\in(0,\infty)$. According to Fatau lemma, by
$G\in\mathcal{C}\subset\mathcal{L}$, we have
$\overline{F}(x)\sim\overline{G}(x)$ and $F\in\mathcal{C}$. To prove (5.6), we
first give two facts.

######  Lemma 5.1.

In Case $i)$, let $Y_{i},i\geq 1$ be ENOD random variables with common
distribution $G\in\mathcal{C}$, then for each $n_{0}\geq 1$, it is holds
uniformly for all $1\leq n\leq n_{0}$ that

$\displaystyle P(\max_{1\leq k\leq n}S^{F}_{k}>x)\sim P(S^{F}_{n}>x)\sim
n\overline{G}(x)\sim n\overline{G}(x-\mu_{G}n)\sim
n\overline{G}(x-\mu_{G}n+\mu_{H}n).$ (5.7)

Further, there is a positive function $n(\cdot)$ such that
$n(x)\uparrow\infty,\ n(x)x^{-1}\to 0$ and (5.7) is holds uniformly for all
$1\leq n\leq n(x)$.

###### Proof.

For each $n_{0}\geq 1$, on the one hand, according to Lemma 2.3 of Wang et al.
[48], by $G\in\mathcal{C}\subset\mathcal{L}\cap\mathcal{D}$, for each
$n_{0}\geq 1$ and all $1\leq n\leq n_{0}$, we have

$\displaystyle P(\max_{1\leq k\leq n}S^{F}_{k}>x)\leq P(S^{G}_{n}>x)\sim
n\overline{G}(x)\sim n\overline{G}(x-\mu_{G}n)\sim
n\overline{G}(x-\mu_{G}n+\mu_{H}n).$ (5.8)

On the other hand, according to Fatou lemma, by
$G\in\mathcal{C}\subset\mathcal{S}$ and (5.8), for each $n_{0}\geq 1$ and all
$1\leq n\leq n_{0}$, we have

$\displaystyle\lim\frac{P(\max_{1\leq k\leq
n}S^{F}_{k}>x)}{n\overline{G}(x)}\geq\lim\int\cdots\int\frac{P\big{(}S^{G}_{n}>x+c\sum_{i=1}^{n}z_{i}\big{)}}{n\overline{G}\big{(}x+c\sum_{i=1}^{n}z_{i}\big{)}}\frac{\overline{G}\big{(}x+c\sum_{i=1}^{n}z_{i}\big{)}}{\overline{G}(x)}$
(5.9) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ P(Z_{1}\in dz_{1},\cdots,Z_{n}\in dz_{n})$ $\displaystyle=$ $\displaystyle
1.$

Therefore, by (5.8) and (5.9), (5.7) holds for each $n_{0}\geq 1$ and all
$1\leq n\leq n_{0}$.

Further, for each $n_{0}=m\geq 1$ and all $1\leq n\leq m$, by (5.7), there is
a $x_{m}\in(0,\infty)$ such that $x_{m}<\max\\{x_{m+1},\ m^{2}\\}$ and when
$x\in[x_{m},x_{m+1})$,

$1-m^{-1}<P\big{(}\max_{1\leq k\leq
n}S^{F}_{k}>x\big{)}\big{(}na(x)\big{)}^{-1}<1+m^{-1},$

where $a(x)$ is one of $\overline{G}(x),\ \overline{G}(x-\mu_{G}n)$ and
$\overline{G}(x-\mu_{G}n+\mu_{H}n)$. Let $x_{0}=0$ and

$n(x)=\sum_{m=1}^{\infty}m\textbf{1}_{\\{x\in[x_{m-1},x_{m}\\}}(x),\ \ \ \
x\in[0,\infty).$

Then $n(x)\uparrow\infty,\ n(x)x^{-1}\to 0$ and (5.7) is holds uniformly for
all $1\leq n\leq n(x)$. $\Box$

######  Lemma 5.2.

In Cases $(i)$, let $Y_{i},i\geq 1$ be ENOD random variables with common
distribution $G\in\mathcal{D}$, and let $Z_{i},\ i\geq 1$ be WOD random
variables with the dominating coefficients $g_{U,H}(n)$ and $g_{L,H}(n),\
n\geq 1$. If there exists some almost increasing function $g(\cdot)$, some
integer $m\geq 1$ and some $\omega\in(0,1)$ such that
$EZ_{1}g(Z_{1})\in(0,\infty)$ and condition (1.7) is satisfied, then for each
$\gamma\in(\mu_{G},\infty)$,

$\displaystyle\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\leq\liminf_{n\to\infty}\inf_{x\in[\gamma
n,\infty)}P\big{(}\max_{1\leq k\leq
n}S^{F}_{k}>x\big{)}\big{(}n\overline{G}(x)\big{)}^{-1}$ (5.10)
$\displaystyle\leq$ $\displaystyle\limsup_{n\to\infty}\sup_{x\in[\gamma
n,\infty)}P\big{(}\max_{1\leq k\leq
n}S^{F}_{k}>x\big{)}\big{(}n\overline{G}(x)\big{)}^{-1}\leq\overline{G}^{-1}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}.$

###### Proof.

On the one hand, we estimate the asymptotic lower bound of $P(\max_{1\leq
k\leq n}S^{F}_{k}>x)$. To this end, for any $\delta\in(0,1)$, we denote

$D(n)=D(n,H,\delta)=\Big{\\{}(1-\delta)\mu_{H}n\leq\sum_{i=1}^{n}Z_{i}\leq(1+\delta)\mu_{H}n\Big{\\}},\
\ \ \ n\geq 1.$

Then according to Proposition 1.6, we have

$\lim_{n\to\infty}P\big{(}D(n)\big{)}=1.$

In Case $(i)$, according to Lemma 3.1, by the proof method in (4.12) and
$\mu_{G}<c\mu_{H}$, for any $\varepsilon\in(0,1)$ and each
$\gamma\in(\mu_{G},\infty)$, there is
$\delta_{0}=\delta_{0}(\varepsilon,c,H)\in(0,1)$ and
$n_{0}=n_{0}(\varepsilon,c,\gamma,G)\geq 1$ such that, when
$\delta\in(0,\delta_{0})$ and $n\geq n_{0}$, it holds uniformly for all
$x\in[\gamma n,\infty)$ that

$\displaystyle P\big{(}\max_{1\leq k\leq n}S^{F}_{k}>x\big{)}\geq
P\Big{(}S_{n}^{G}>x+c\sum_{i=1}^{n}Z_{i},D(n)\Big{)}$ (5.11)
$\displaystyle\geq$ $\displaystyle
P\big{(}S_{n}^{G}>x+c(1+\delta)\mu_{H}n\big{)}P\big{(}D(n)\big{)}$
$\displaystyle>$
$\displaystyle(1-\varepsilon)n\overline{G}\big{(}x+c(1+\delta)\mu_{H}n-\mu_{G}n\big{)}$
$\displaystyle>$
$\displaystyle(1-\varepsilon)^{2}n\overline{G}(x+c\mu_{H}n-\mu_{G}n)$
$\displaystyle=$
$\displaystyle(1-\varepsilon)^{2}n\overline{G}\big{(}x(1+(c\mu_{H}n-\mu_{G}n)x^{-1})\big{)}$
$\displaystyle\geq$
$\displaystyle(1-\varepsilon)^{2}n\overline{G}\big{(}x(1+(c\mu_{H}n-\mu_{G}n)(\gamma
n)^{-1})\big{)}$ $\displaystyle\geq$
$\displaystyle(1-\varepsilon)^{3}\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}n\overline{G}(x).$

On the other hand, according to Lemma 3.1, by $G\in\mathcal{D}$, for any
$\varepsilon\in(0,1)$ and each $\gamma\in(\mu_{G},\infty)$, there is
$n_{0}=n_{0}(\varepsilon,c,\gamma,G)\geq 1$, when $n\geq n_{0}$, it holds
uniformly for all $x\in[\gamma n,\infty)$ that,

$\displaystyle P(\max_{1\leq k\leq n}S^{F}_{k}>x)\leq
P\big{(}S_{n}^{G}>x\big{)}$ (5.12) $\displaystyle<$
$\displaystyle(1+\varepsilon)n\overline{G}(x-\mu_{G}n)$ $\displaystyle=$
$\displaystyle(1+\varepsilon)n\overline{G}\big{(}x(1-\mu_{G}nx^{-1})\big{)}$
$\displaystyle\leq$
$\displaystyle(1+\varepsilon)n\overline{G}\big{(}x(1-\mu_{G}\gamma^{-1})\big{)}$
$\displaystyle\leq$
$\displaystyle(1+\varepsilon)^{2}\overline{G}_{*}^{-1}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}n\overline{G}(x).$

Combining (5.11), (5.12) and the arbitrariness of $\varepsilon$, we know that
(5.10) holds. $\Box$

######  Lemma 5.3.

In a nonstandard renewal risk model, let $Z_{i},\ i\geq 1$ be WLOD random
variables satisfying (1.3) and be independent of a nonnegative random variable
$\tau$. If $G\in\mathcal{D}$ and (5.5) is satisfied, then

$\displaystyle\lim P\big{(}N(\tau)>x\big{)}\overline{G}^{-1}(x)=0.$ (5.13)

###### Proof.

According to Proposition 1.4, by (1.3), we know that for any constant
$\delta\in(0,\infty)$, there exist three positive finite constants $t_{0}$,
$r=r(H,t_{0})$ and $C=C(r,\delta,H)$ such that $Ee^{rN(t_{0})}\in(0,\infty)$
and

$\displaystyle\sup_{t\in[0,\infty)}Ee^{rN(t)}\textbf{1}_{\\{N(t)>\frac{(1+\delta)t}{\mu_{H}}\\}}\leq\max\big{\\{}\sup_{t\in[0,t_{0}]}Ee^{rN(t)}\textbf{1}_{\\{N(t)>\frac{(1+\delta)t}{\mu_{H}}\\}},\
\sup_{t\in(t_{0},\infty)}Ee^{rN(t)}\textbf{1}_{\\{N(t)>\frac{(1+\delta)t}{\mu_{H}}\\}}\big{\\}}$
(5.14) $\displaystyle\leq$ $\displaystyle\max\big{\\{}Ee^{rN(t_{0})},\
\sup_{t\in(t_{0},\infty)}Ee^{rN(t)}\textbf{1}_{\\{N(t)>\frac{(1+\delta)t}{\mu_{H}}\\}}\big{\\}}$
$\displaystyle\leq$ $\displaystyle C.$

Then we split

$\displaystyle
P\big{(}N(\tau)>x\big{)}=\Big{(}\int_{0-}^{\mu_{H}(1+\delta)^{-1}x}+\int_{\mu_{H}(1+\delta)^{-1}x}^{\infty}\Big{)}P\big{(}N(t)>x\big{)}P(\tau\in
dt)$ (5.15) $\displaystyle=$ $\displaystyle I_{1}(x)+I_{2}(x),\ \ \ \ \ \ \ \
x\in[0,\infty).$

For $I_{2}(x)$, by (5.5) and $G\in\mathcal{D}$, we have

$\displaystyle I_{2}(x)\leq
P\big{(}\tau>\mu_{H}(1+\delta)^{-1}x\big{)}=o\big{(}\overline{G}(\mu_{H}(1+\delta)^{-1}x)\big{)}=o\big{(}\overline{G}(x)\big{)}.$
(5.16)

For $I_{1}(x)$, according to Markov inequality and Proposition 1.2 $(iii)$, by
(5.14), we have

$\displaystyle I_{1}(x)\leq
e^{-rx}\int_{0-}^{\mu_{H}(1+\delta)^{-1}x}Ee^{rN(t)}\textbf{1}_{\\{N(t)>x)\\}}P(\tau\in
dt)$ (5.17) $\displaystyle\leq$ $\displaystyle
e^{-rx}\int_{0-}^{\mu_{H}(1+\delta)^{-1}x}\sup_{t\in[0,\infty)}Ee^{rN(t)}\textbf{1}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}P(\tau\in
dt)\overline{G}^{-1}(x)\overline{G}(x)$ $\displaystyle\leq$ $\displaystyle
Ce^{-rx}\overline{G}^{-1}(x)\overline{G}(x)$ $\displaystyle=$ $\displaystyle
o\big{(}\overline{G}(x)\big{)},$

that is (5.13) holds. $\Box$

Now, we continue to prove the theorem. By (5.13) in Lemma 5.3 and
$\mu_{G}\in(0,\infty)$, we have $EN(\tau)\in(0,\infty)$. Thus,

$E\lambda(\tau)=\int_{0-}^{\infty}\lambda(t)P(\tau\in
dt)=\int_{0-}^{\infty}EN(t)P(\tau\in dt)=EEN(\tau)=EN(\tau)<\infty.$

In order to prove (5.6), we split

$\displaystyle\psi(x;\tau)=\Big{(}\int_{0-}^{x}+\int_{x}^{\infty}\Big{)}P\big{(}\max_{1\leq
n\leq N(t)}S^{F}_{n}>x\big{)}P(\tau\in dt)$ (5.18) $\displaystyle=$
$\displaystyle I_{1}(x)+I_{2}(x),\ \ \ \ \ \ \ \ \ x\in(0,\infty).$

We first deal with $I_{2}(x)$. By (5.5), we know that

$\displaystyle I_{2}(x)\leq P(\tau>x)=o\big{(}\overline{G}(x)\big{)}.$ (5.19)

Next, we deal with $I_{1}(x)$. For any $\varepsilon\in(0,1)$ and
$\gamma\in(\mu_{G},\infty)$, by (5.10) in Lemma 5.2, there exists an integer
$n_{0}=n_{0}(\varepsilon,G)\geq 1$ such that, when $n>n_{0}$, it holds that

$\displaystyle(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\leq\inf_{x\in[\gamma\mu_{G}n,\infty)}P\big{(}\max_{1\leq
k\leq n}S^{F}_{k}>x\big{)}\big{(}n\overline{G}(x)\big{)}^{-1}$ (5.20)
$\displaystyle\leq$
$\displaystyle\sup_{x\in[\gamma\mu_{G}n,\infty)}P\big{(}\max_{1\leq k\leq
n}S^{F}_{k}>x\big{)}\big{(}n\overline{G}(x)\big{)}^{-1}$ $\displaystyle\leq$
$\displaystyle(1+\varepsilon)\overline{G}^{-1}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}.$

And by $G\in\mathcal{C}\subset\mathcal{S}$, (5.7) in Lemma 5.1 and
$EN(\tau)<\infty$, for the above $\varepsilon$ and $n_{0}$, there exists
$x_{0}=x_{0}(n_{0},\varepsilon)\in(0,\infty)$ large enough such that

$\displaystyle
1-\varepsilon\leq\inf_{x\in[x_{0},\infty)}\frac{P\big{(}\max_{1\leq k\leq
n}S^{F}_{k}>x\big{)}}{n\overline{G}(x)}\leq\sup_{x\in[x_{0},\infty)}\frac{P\big{(}\max_{1\leq
k\leq n}S^{F}_{k}>x\big{)}}{n\overline{G}(x)}\leq 1+\varepsilon,$ (5.21)

$n\leq n_{0}$ and

$\displaystyle EN(\tau)\textbf{1}_{\\{N(\tau)\geq\min\\{N(x),\
\gamma^{-1}\mu_{G}^{-1}x\\}\\}}<\varepsilon,\ \ \ x\geq x_{0}.$ (5.22)

Then we again split

$\displaystyle
I_{1}(x)=\int_{0-}^{x}\Big{(}\sum_{n=1}^{n_{0}}+\sum_{n=n_{0}+1}^{\infty}\Big{)}P\big{(}\max_{1\leq
k\leq n}S^{F}_{k}>x,N(t)=n\big{)}P(\tau\in dt)$ (5.23) $\displaystyle=$
$\displaystyle I_{11}(x)+I_{12}(x).$

For $I_{11}(x)$, by (5.21), when $x\in[x_{0},\infty)$,

$\displaystyle
I_{11}(x)\leq(1+\varepsilon)\overline{G}(x)\int_{0-}^{x}\sum_{n=1}^{n_{0}}nP\big{(}N(t)=n\big{)}P(\tau\in
dt)$ (5.24) $\displaystyle\leq$
$\displaystyle(1+\varepsilon)EN(\tau)\textbf{1}_{\\{N(\tau)\leq
n_{0}\\}}\overline{G}(x).$

It is similar to get that

$\displaystyle
I_{11}(x)\geq(1-\varepsilon)\overline{G}(x)\int_{0-}^{x}\sum_{n=1}^{n_{0}}nP\big{(}N(t)=n\big{)}P(\tau\in
dt)$ (5.25) $\displaystyle\geq$
$\displaystyle(1-\varepsilon)EN(\tau)\textbf{1}_{\\{N(\tau)\leq
n_{0}\\}}\overline{G}(x).$

For $I_{12}(x)$, we further split it as follows:

$\displaystyle
I_{12}(x)=\int_{0-}^{\infty}\sum_{n=n_{0}+1}^{\infty}\big{(}\textbf{1}_{\\{t:\
x\geq\max\\{t,\ \gamma\mu_{G}n)\\}}+\textbf{1}_{\\{t:\ 0\leq t\leq
x<\gamma\mu_{G}n\\}}\big{)}$ (5.26) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \cdot P\big{(}\max_{1\leq k\leq
n}S_{k}^{F}>x,N(t)=n\big{)}P(\tau\in dt)$ $\displaystyle=$ $\displaystyle
I_{121}(x)+I_{122}(x).$

By (5.20), we have

$\displaystyle
I_{121}(x)\leq\frac{(1+\varepsilon)\overline{G}(x)}{\overline{G}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}}\int_{0-}^{\infty}\sum_{n=n_{0}+1}^{\infty}nP\big{(}N(t)=n\big{)}\textbf{1}_{\\{t:\
x\in[\max\\{t,\gamma\mu_{G}n)\\},\infty)}(t)P(\tau\in dt)$ (5.27)
$\displaystyle\leq$
$\displaystyle\frac{(1+\varepsilon)\overline{G}(x)}{\overline{G}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}}\int_{0-}^{\infty}\sum_{n=n_{0}+1}^{\infty}nP\big{(}N(t)=n\big{)}P(\tau\in
dt)$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{(1+\varepsilon)\overline{G}(x)}{\overline{G}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}}\int_{0-}^{\infty}\sum_{n=n_{0}+1}^{\infty}nP\big{(}N(t)=n\big{)}\textbf{1}_{\\{t:\
0\leq t\leq x<\gamma\mu_{G}n\\}}(t)P(\tau\in dt)$ $\displaystyle=$
$\displaystyle\frac{(1+\varepsilon)\overline{G}(x)}{\overline{G}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}}EN(\tau)\textbf{1}_{\\{N(\tau)>n_{0}\\}}-\frac{(1+\varepsilon)\overline{G}(x)}{\overline{G}_{*}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}}I^{*}_{122}(x),$
$\displaystyle
I_{121}(x)\geq(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)$
(5.28) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot\int_{0-}^{\infty}\sum_{n=n_{0}+1}^{\infty}nP\big{(}N(t)=n\big{)}\textbf{1}_{\\{t:\
x\in[\max\\{t,\gamma\mu_{G}n)\\},\infty)}(t)P(\tau\in dt)$ $\displaystyle=$
$\displaystyle(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)EN(\tau)\textbf{1}_{\\{N(\tau)>n_{0},x\in[\tau,\infty)\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)I^{*}_{122}(x)$
$\displaystyle\geq$
$\displaystyle(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)EN(\tau)\textbf{1}_{\\{N(\tau)>n_{0}\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)EN(\tau)\textbf{1}_{\\{x\in(0,\tau)\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(1-\varepsilon)\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\overline{G}(x)I^{*}_{122}(x)$

and

$\displaystyle
I_{122}(x)\leq\int_{0-}^{\infty}P(N(t)>\gamma^{-1}\mu_{G}^{-1}x)P(\tau\in
dt)=P(N(\tau)>\gamma^{-1}\mu_{G}^{-1}x)=o(\overline{G}(x)).$ (5.29)

Here, by (5.13) in Lemma 5.3, (5.22) and $G\in\mathcal{C}\subset\mathcal{D}$,
for the above $\varepsilon$, $x_{0}$ and $n_{0}$,

$\displaystyle I^{*}_{122}(x)\leq
EN(\tau)\textbf{1}_{\\{N(\tau)>\gamma^{-1}\mu_{G}^{-1}x\\}}<\varepsilon.$
(5.30)

Combining (5.18), (5.19), (5.22) and (5.23)-(5.30), by the the arbitrariness
of $\varepsilon$, $EN(\tau)\in(0,\infty)$, $\\{\tau>x\\}\subset\\{N(\tau)\geq
N(x)\\}$ and the fact that

$\overline{G}_{*}\big{(}1+(c\mu_{H}-\mu_{G})\gamma^{-1}\big{)}\leq 1\ \ \
\text{and}\ \ \
\overline{G}_{*}^{-1}\big{(}(1-\mu_{G}\gamma^{-1})^{-1}\big{)}\geq 1,$

we have

$\displaystyle\overline{G}_{*}\Big{(}1+\frac{c\mu_{H}-\mu_{G}}{\gamma}\Big{)}\leq\liminf\frac{\psi(x;\tau)}{E\lambda(\tau)\overline{G}(x)}\leq\limsup\frac{\psi(x;\tau)}{E\lambda(\tau)\overline{G}(x)}\leq\frac{1}{\overline{G}_{*}\big{(}(1-\frac{\mu_{G}}{\gamma})^{-1}\big{)}}.$
(5.31)

Let $\gamma\uparrow\infty$, then by (5.31) and $G\in\mathcal{C}$, we know that
(5.6) holds. $\Box$

At the end of this subsection, we give a sufficient condition for (5.5).

######  Proposition 5.1.

In the above nonstandard renewal risk model Case $(i)$, let $Z_{i},i\geq 1$ be
WLOD random variables satisfying all conditions in Proposition 1.6. If
$G\in\mathcal{D}$ and $EN^{p}(\tau)\in(0,\infty)$ for some
$p\in(J_{G}^{+},\infty)$, then (5.5) holds.

###### Proof.

According to Proposition 1.2 $(iii)$, for each $p\in(J_{G}^{+},\infty)$, by
$G\in\mathcal{D}$, we have

$\lim x^{p}\overline{G}(x)=\infty.$

And by $EN^{p}(\tau)\in(0,\infty)$, we have

$\lim x^{p}P\big{(}N(\tau)>x\big{)}=0.$

Combined with the above two asymptotic expressions, we know that

$P\big{(}N(\tau)>x\big{)}=o\big{(}\overline{G}(x)\big{)}.$

By the fact, we have

$\displaystyle
P(\tau>x)=P\big{(}\tau>x,N(\tau)>2^{-1}\mu_{H}^{-1}x\big{)}+P\big{(}\tau>x,N(\tau)\leq
2^{-1}\mu_{H}^{-1}x\big{)}$ $\displaystyle\leq$ $\displaystyle
P\big{(}N(\tau)>2^{-1}\mu_{H}^{-1}x\big{)}+\int_{x}^{\infty}P\big{(}N(t)\leq
2^{-1}\mu_{H}^{-1}x\big{)}P(\tau\in dt)$ $\displaystyle\leq$ $\displaystyle
o\big{(}\overline{G}(x)\big{)}+P\big{(}N(x)\leq
2^{-1}\mu_{H}^{-1}x\big{)}P(\tau>x).$

Therefore, for $x$ large enough, it holds that

$P(\tau>x)\leq
2P(\tau>x)P\big{(}N(x)>2^{-1}\mu_{H}^{-1}x\big{)}=o\big{(}\overline{G}(x)\big{)}$

according to Proposition 1.6. $\Box$

###  5.3 Precise large deviation of finite-times ruin probability

######  Theorem 5.3.

$(i)$ Under conditions as Theorem 2.1 $(i)$, if

$\displaystyle\overline{H}(x)=o\big{(}\overline{G}(x)\big{)},$ (5.32)

then for each $\gamma\in(\mu_{G_{1}},\infty)$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma\mu_{H}^{-1}t,\infty)}\big{|}\psi(x;\tau)\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G_{1}}\mu_{H}^{-1}t+ct)\big{)}^{-1}-1\big{|}=0.$
(5.33)

$(ii)$ Under conditions as Theorem 2.2, if (5.32) is satisfied, then for each
pair $\mu_{G}<\gamma<\Gamma<\infty$,

$\displaystyle\lim\limits_{t\to\infty}\sup\limits_{x\in[\gamma t,\Gamma
t]}\big{|}\psi(x;\tau)\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{G_{1}}\mu_{H}^{-1}t+ct)\big{)}^{-1}-1\big{|}=0.$
(5.34)

$(iii)$ Under conditions as Theorem 2.3, if (5.32) is satisfied, then (5.34)
still holds for each pair $\mu_{G}<\gamma<\Gamma<\infty$.

###### Proof.

$(i)$ On the one hand, according to Theorem 2.1, for each
$\gamma\in[\mu_{G},\infty)$, it holds uniformly $x\in[\gamma t,\infty)$ that

$\displaystyle\psi(x;t)\geq
P\Big{(}\sum_{i=1}^{N(t)}(Y_{i}-cZ_{i})>x\Big{)}\sim\mu_{H}^{-1}t\overline{G}(x-\mu_{G}\mu_{H}^{-1}t+ct).$
(5.35)

On the other hand, we take any positive function $m(\cdot)$ such that
$m(t)\uparrow\infty$ and $m(t)\downarrow 0$ as $t\to\infty$. By the above
$m(\cdot)$ and any $\delta\in(0,1)$, we sprite

$\displaystyle\psi(x;t)=\Big{(}\sum_{m=1}^{\lfloor
m(t)\rfloor}+\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}+\sum_{m=\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor+1}^{\infty}\Big{)}P\Big{(}\max_{1\leq
n\leq m}\sum_{i=1}^{n}(Y_{i}-cZ_{i})>x,N(t)=m\Big{)}$ (5.36) $\displaystyle=$
$\displaystyle\psi_{1}(x;t)+\psi_{2}(x;t)+\psi_{3}(x;t).$

First, according to Lemma 3.1 and Lemma 5.1, by $m(t)=o(t)$ and
$\mu_{G}<c\mu_{H}$, when $t\to\infty$, it holts uniformly for all $x\in[\gamma
t,\infty)$ that

$\displaystyle\psi_{1}(x;t)\leq\sum_{m=1}^{\lfloor
m(t)\rfloor}P\Big{(}\sum_{i=1}^{\lfloor
m(t)\rfloor}Y_{i}>x\Big{)}P\big{(}N(t)=m\big{)}$ (5.37) $\displaystyle\sim$
$\displaystyle m(t)\overline{G}\big{(}x-\mu_{G}m(t)\big{)}\sum_{m=1}^{\lfloor
m(t)\rfloor}P\big{(}N(t)=m\big{)}\Big{)}$ $\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{H}^{-1}\mu_{G}t+ct)\big{)}.$

Next, for each $\gamma\in[\mu_{G},\infty)$, according to Lemma 3.1, by
$\mu_{G}<c\mu_{H}$, when $t\to\infty$, it holts uniformly for all $x\in[\gamma
t,\infty)$ that

$\displaystyle\psi_{3}(x;t)\leq\sum_{m=\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor+1}^{\infty}P\Big{(}\sum_{i=1}^{m}Y_{i}>x\Big{)}P\big{(}N(t)=m\big{)}$
(5.38) $\displaystyle\sim$
$\displaystyle\sum_{m=\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor+1}^{\infty}m\overline{G}(x-m\mu_{G})P\big{(}N(t)=m\big{)}$
$\displaystyle\leq$
$\displaystyle\overline{G}(x-(1+\delta)\mu_{H}^{-1}t\mu_{G})EN(t)\textbf{1}_{\\{N(t)>(1+\delta)\mu_{H}^{-1}t\\}}(t)$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{H}^{-1}\mu_{G}t+ct)\big{)}.$

Finally, for each $\gamma\in[\mu_{G},\infty)$, we take any $\delta\in(0,1)$
such that $\gamma>(1-\delta)\mu_{G}$. Then we further sprite

$\displaystyle\psi_{2}(x;t)\leq\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}\bigg{(}P\Big{(}\max_{1\leq
n\leq\lfloor m(t)\rfloor}\sum_{i=1}^{n}(Y_{i}-cZ_{i})>x,N(t)=m\Big{)}$ (5.39)
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ +P\Big{(}\max_{\lfloor
m(t)\rfloor+1\leq n\leq
m}\sum_{i=1}^{n}(Y_{i}-cZ_{i})>x,N(t)=m\Big{)}\bigg{)}$ $\displaystyle=$
$\displaystyle\psi_{21}(x;t)+\psi_{22}(x;t).$

According to Lemma 3.1, when $t\to\infty$, it holds uniformly for all
$x\in[\gamma t,\infty]$ that

$\displaystyle\psi_{21}(x;t)\leq\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}P\Big{(}\sum_{i=1}^{\lfloor
m(t)\rfloor}Y_{i}>x\Big{)}P\big{(}N(t)=m\big{)}$ (5.40) $\displaystyle\sim$
$\displaystyle
m(t)\overline{G}\big{(}x-\mu_{G}m(t)\big{)}E\textbf{1}_{\\{N(t)>m(t)\\}}$
$\displaystyle=$ $\displaystyle
o\big{(}\mu_{H}^{-1}t\overline{G}(x-\mu_{H}^{-1}\mu_{G}t+ct)\big{)}.$

To deal with $\psi_{22}(x;t)$, let

$D_{n}=D_{n}(H,t,\delta)=\Big{\\{}(1-\delta)\mu_{H}n\leq\sum_{i=1}^{n}Z_{i}\leq(1+\delta)\mu_{H}n\Big{\\}},\
\ \ \ \ n\geq 1.$

By $D_{n},\ n\geq 1$, we continue sprite

$\displaystyle\psi_{22}(x;t)=\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}P\Big{(}\max_{\lfloor
m(t)\rfloor+1\leq n\leq
m}\sum_{i=1}^{n}(Y_{i}-cZ_{i})>x,N(t)=m,\bigcap_{n=\lfloor(1+\delta)t\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}D_{n}\Big{)}$
(5.41) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}P\Big{(}\max_{\lfloor
m(t)\rfloor+1\leq n\leq
m}\sum_{i=1}^{n}(Y_{i}-cZ_{i})>x,N(t)=m,\bigcup_{n=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}D_{n}^{c}\Big{)}$
$\displaystyle=$ $\displaystyle\psi_{221}(x;t)+\psi_{222}(x;t).$

According to Lemma 3.1, when $t\to\infty$ and then $\delta\downarrow 0$, it
holds uniformly for all $x\in[\gamma t,\infty]$ that

$\displaystyle\psi_{221}(x;t)\leq\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}P\Big{(}\max_{\lfloor
m(t)\rfloor+1\leq n\leq
m}\sum_{i=1}^{n}\big{(}Y_{i}-c(1-\delta)\mu_{H}\big{)}>x\Big{)}P\big{(}N(t)=m\big{)}$
(5.42) $\displaystyle\leq$ $\displaystyle\sum_{m=\lfloor
m(t)\rfloor+1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}P\Big{(}\sum_{i=1}^{\lfloor(1+\delta)\mu_{H}^{-1}t\rfloor}\big{(}Y_{i}-c(1-\delta)\mu_{H}\big{)}>x\Big{)}P\big{(}N(t)=m\big{)}$
$\displaystyle\lesssim$
$\displaystyle(1+\delta)\mu_{H}^{-1}t\overline{G}\big{(}x-(1+\delta)\mu_{H}^{-1}t(\mu_{G}-c(1-\delta)\mu_{H})\big{)}$
$\displaystyle\sim$
$\displaystyle\mu_{H}^{-1}t\overline{G}\big{(}x-\mu_{H}^{-1}\mu_{G}t+ct\big{)}.$

By (4.2),
$\psi_{222}(x;t)=o\Big{(}\mu_{H}^{-1}t\overline{G}\big{(}x-\mu_{H}^{-1}\mu_{G}t\big{)}\Big{)}=o\Big{(}\mu_{H}^{-1}t\overline{G}\big{(}x-\mu_{H}^{-1}\mu_{G}t+ct\big{)}\Big{)}.$

Combining (5.35) and (5.36), we know that (5.34) holds. $\Box$

## References

  * [1] Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scandinavian Actuarial Journal, 31-57.
  * [2] Bingham N. H., Goldie C. M. and Teugels, J. Z. (1987). Regular Variation. Cambridge University Press.
  * [3] Block H. W., Savits T. H. and Shaked M. (1982). Some concept of negative dependence. The Annals of Probability, 10, 765-772.
  * [4] Chen W., Wang Y. and Cheng D. (2016). An inequality of widely dependent random variables and its applications. Lithuanian Mathematical Journal, 56(1), 16-31.
  * [5] Chen Y., Yuen K.C and Ng K.W. (2011). Precise large deviations of random sums in presence of negative dependence and consistent variation. Methodology and Computing in Applied Probability, 13, 821-833.
  * [6] Chen, Y., White T. and Yuen K. (2021). Precise large deviations of aggregate claims with arbitrary dependence between claim sizes and waiting times. Insurance: Mathematics and Economics, 97, 1-6.
  * [7] Cheng, F. and Li, M. (2016). Local precise large and moderate deviations for sums of independent random variables. Chinese Annals of Mathematics, Series B, 37, 753-766.
  * [8] Chistyakov. V. P. (1964). A theorem on sums of independent positive random variables and its application to branching processes. Probability Theory and its Applications, 9, 640-648.
  * [9] Cline D. B. H. and Hsing T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Corpus Christi: Texas A$\&$M University.
  * [10] Cline D. B. H. and Samorodnitsky G. (1994). Subexponentiality of the product of independent random variables. Stochastic Process and Thier Applications, 49, 75-98.
  * [11] Denisov D., Dieker A. B. and Shneer, V. (2008). Large deviations for random walks under subexponentiality: the big domain. The Annals of Probability, 36(5), 1946-1991.
  * [12] Doney, R. A. (1989). A large deviation local limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 105, 575-577.
  * [13] Ebrahimi N. and Ghosh M. (1981). Multivariate negative dependence. Communications in Statistics, 10, 307-337.
  * [14] Embrechts P., Klüppelberg C. and Mikosch T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  * [15] Foss S., Korshunov D. and Zachry S. (2013). An Introduction to Heavy-tailed and Subexponential Distributions. Springer, Second Edition.
  * [16] Feller W. (1971). An Introduction to Probability Theory and its Applications, 2nd edn. vol. 2, Wiley, New York.
  * [17] Gao Q., Lin J. and Liu X. (2023). Large deviations of aggregate amount of claims in compound risk model with arbitrary dependence between claim sizes and waiting times. Statistics and Probability Letters, 197, 109809.
  * [18] Heyde C. C. (1967a). A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 7, 303-308.
  * [19] Heyde C. C. (1967b). On large deviation problem for sums of random variables which are not attracted to the normal law. The Annals of Mathematical Statistics, 38, 1575-1578.
  * [20] Heyde C. C. (1968). On large deviation probabilities in the case of attraction to a nonnormal stable law. Scankhyā A, 30, 253-258.
  * [21] Joag-Dev K. and Proschan F. (1983). Negative association of random variables with aaapplitions. The Annals of Statistics, 11, 286-295.
  * [22] Klüppelberg C. (1988). Subexponential distributions and integrated tails. Jounal of Appllied Probability, 25, 132-141.
  * [23] Klüppelberg C. and Mikosch T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. Jounal of Appllied Probability, 34, 293-308.
  * [24] Klüppelberg, C. and Stadtimuller, U. (1998). Ruin probability in the presence of heavy-tails and interest rates. Scandinavian Actuarial Journal 1, 49-58.
  * [25] Konstantinides D. and Mikosch T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. The Annals of Probability, 33, 1992-2035.
  * [26] Lehmann E. L. (1966). Some concepts of dependence. Annals of Mathematical Statistics, 43, 1137-1153.
  * [27] Leipus, R. and Šiaulys, J. (2007). Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes. Insurance: Mathematics and Economics, 40, 498-508.
  * [28] Lin, J. (2008). A one-sided large deviation local limits theorem, Statistics and Probability Letters, 78, 2679-2684.
  * [29] Liu L. (2009). Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters, 79, 1290-1298.
  * [30] Liu X., Gao Q. and Wang Y. (2012). A note on a dependent risk model with constant interest rate. Statistics and Probability Letters, 82, 707-712.
  * [31] Mikosch T. and Wintenberger O. (2013). Precise large deviations for dependent regularly varying sequences. Probability Theory and Related Fields, 156, 851-887.
  * [32] Mikosch T. and Nagaev A.V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes, 1, 81-110.
  * [33] Nagaev A. V. (1969a). Integral limit theorems for large deviations when Cramér’s condition is not fulfilled.I. Probability Theory and its Applications, 14, 51-64.
  * [34] Nagaev A. V.(1969b). Integral limit theorems for large deviations when Cramér’s condition is not fulfilled.II. Probability Theory and its Applications, 14, 193-208.
  * [35] Nagaev A. V. (1973). Large deviations for sums of independent random variables. In Trans. Sixth Prague Conf. Inf. Theory Statist. Decision Functions Random Process, Academia, 657-674.
  * [36] Nagaev A. V. (1979). Large deviations of sums of independent random variables. The Annals of Probability, 7, 745-789.
  * [37] Ng K. W., Tang Q., Yan J. and Yang H. (2004). Precise large deviations for sums of random variables with consistently varying tails. Jounal of Appllied Probability, 41, 93-107.
  * [38] Tang Q. (2004). Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stochastic Models, 20, 281-297.
  * [39] Tang Q. (2006). Insensitivity to negative dependence of the asymptotic behavior of precise deviations. Electronic Journal of Probability, 11, 107-120.
  * [40] Tang Q. Su C. Jiang T. and Zhang J.S. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statistics and Probability Letters, 52, 91-100.
  * [41] Tang Q. and Tsitsiashvili G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process and Thier Applications, 108, 299-325.
  * [42] Wang Y. and Cheng D. (2011). Basic renewal theorems for a random walk with widely dependent increments and their applications. Journal of Mathematical Analysis and Applications, 384, 597-606.
  * [43] Wang Y. and Cheng D. (2020). Elementary renewal theorems for widely dependent random variables with applications to precise large deviations. Communications in Statistics-Theory and Methods, 49 (14), 3352-3374.
  * [44] Wang Y., Cui Z., Wang K. and Ma X. (2012). Uniform asymptotics of the finite-time ruin probability for all times. Journal of Mathematical Analysis and Applications, 390, 208-223.
  * [45] Wang Y., Gao Q., Wang K. and Liu X. (2009). Random time ruin probability for the renewal risk model with heavy-tailed claims. Jonarnal of Industrial and Management Optimization, 5(4), 2009, 719-736.
  * [46] Wang S. and Wang W. (2007). Precise large deviations for sums of random variables with consistently varying tails in multi-risk models. Jounal of Appllied Probability, 44, 4, 889-900.
  * [47] Wang Y., Wang K. and Cheng D. (2006). Precise large deviations for sums of negatively associated random variables with common dominatedly varying tails. Acta Mathematica Sinica, English Serie, 22 (6), 1725-1734.
  * [48] Wang K., Wang Y. and Gao Q. (2013). Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodology and Computing in Applied Probability, 15, 109-124.
  * [49] Wang K., Yang Y. and Lin J. (2012). Precise large deviations for widely orthant dependent random variables with dominatedly varying tails. Frontiers of Mathematics in China, 7, 919-932.
  * [50] Yang Y., Leipus R. and $\breve{S}$iaulys, J. Local presice deviations for sums of random variables with O-reqularly varying densities. Statistics and Probability Letters, 80, 2010, 1559-1567.