© Wuhan University 2023

0 Introduction

In this section we consider a one-dimensional risk model, in which the surplus at time $t\ge \mathrm{0}$ is described as

$U(t)=x+ct-{\displaystyle \sum _{i=\mathrm{1}}^{N(t)}}{X}_i$(1)

where $x\ge \mathrm{0}$ is the initial surplus, $c>\mathrm{0}$ is the constant premium rate and the claim size $\left\{X_i,i\ge \mathrm{1}\right\}$ are independent, identically distributed (i.i.d.) and nonnegative random variables with common distribution $F$ and finite mean. $\left\{{\tau }_i,i\ge \mathrm{1}\right\}$ are the claim-arrival times, which constitute the claim-number process

$N(t)=\mathrm{s}\mathrm{u}\mathrm{p}\{i\ge \mathrm{0}:{\tau }_i\le t\},t\ge \mathrm{0}$

with a finite mean function $\lambda (t)=E(N(t))$, $t\ge \mathrm{0}$, where $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{\varnothing }=\mathrm{0}$ and ${\tau }_{\mathrm{0}}=\mathrm{0}$ by convention. The nonnegative random variables $\{{\theta }_i={\tau }_i-{\tau }_{i-\mathrm{1}},i\ge \mathrm{1}\}$ are the claim inter-arrival times, which are independent of $\left\{X_i,i\ge \mathrm{1}\right\}$. For the risk model (1), the finite-time ruin probability up to time $t\ge \mathrm{0}$ is defined as

$\psi (x,t)=P\left(\underset{\mathrm{0}\le s\le t}{\mathrm{i}\mathrm{n}\mathrm{f}}U(s)<\mathrm{0}|U(\mathrm{0})=x\right)$(2)

The risk model (1) has been widely studied and results under various conditions are presented. For the uniform asymptotics of the finite-time ruin probability $\psi (x,t)$ as $x\to \mathrm{\infty }$, when $\left\{{\theta }_i,i\ge \mathrm{1}\right\}$ are i.i.d., Tang ^[1] investigated the case that the claim sizes have consistently-varying-tailed distributions and obtained the asymptotics of $\psi (x,t)$ holds uniformly for $t\in \mathrm{\Lambda }=\{t\ge \mathrm{0}:\lambda (t)>\mathrm{0}\}$. In the case where the distributions of the claim sizes are from a subclass of subexponential distribution class, Leipus and Šiaulys ^[2] presented the asymptotics of $\psi (x,t)$ holds uniformly for $t\in [f(x),\gamma x]$, where $f(x)$ is an infinitely increasing function and $\gamma >\mathrm{0}$ is a constant. Leipus and Šiaulys ^[3] and Kočetova et al ^[4] considered the claim sizes have strong subexponential distributions and showed the asymptotics of $\psi (x,t)$ holds uniformly for $t\in [f(x),\mathrm{\infty })$. Yang et al^[5] and Wang et al^[6] improved the above results by considering the dependent $\left\{{\theta }_i,i\ge \mathrm{1}\right\}$. Chen et al ^[7] established a two-dimensional risk model for (1) and obtained some corresponding results for i.i.d. $\left\{{\theta }_i,i\ge \mathrm{1}\right\}$. Chen et al ^[8] extended the results of Chen et al ^[7] by considering the dependent $\left\{{\theta }_i,i\ge \mathrm{1}\right\}$.

In the above literatures, they mainly considered the claim inter-arrival times $\left\{{\theta }_i,i\ge \mathrm{1}\right\}$ are i.i.d or have some dependence structures. Few articles have studied the claim-number process is non-stationary. In fact, a non-stationary claim-number process may be more practical. Stabile and Torrisi ^[9] derived the infinite and finite time ruin probabilities for the risk model with a non-stationary Hawkes process and light-tailed claim sizes. Recently, Refs.[10,11] considered the claim-number processes may not be stationary and ergodic and satisfy the large deviations principle (LDP for short). A family of probability measures $\{{\mu }_t{\}}_{t\in (\mathrm{0},\mathrm{\infty })}$ on a Hausdorff topological space $(M,{\mathit{ℱ}}_M)$ satisfies the LDP with rate function $I:M\to [\mathrm{0},\mathrm{\infty })$, if $I$ is a lower semi-continuous function and the following inequalities hold for every Borel set $B$:

$\begin{array}{l}-\underset{x\in B^o}{\mathrm{i}\mathrm{n}\mathrm{f}}I(x)\le \underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{\mathrm{1}}{t}\mathrm{l}\mathrm{o}\mathrm{g}{\mu }_t(B)\\ \le \underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\mathrm{1}}{t}\mathrm{l}\mathrm{o}\mathrm{g}{\mu }_t(B)\le -\underset{x\in \overline{B}}{\mathrm{i}\mathrm{n}\mathrm{f}}I(x),\end{array}$

where $B^o$ and $\overline{B}$ denote the interior and closure of $B$, respectively, see, e.g., Dembo et al ^[12] and Bordenave et al^[13].

This section still considers the claim-number process $\left\{N(t),t\ge \mathrm{0}\right\}$ satisfying the LDP and investigates the uniform asymptotics of the finite-time ruin probability $\psi (x,t)$ for the risk model (1). Section 1 presents the main results after introducing necessary preliminaries and the proofs of the main results are given. Section 2 studies a two-dimensional risk model and investigates a kind of finite-time ruin probability by using the results of Section 1.

1 Preliminaries and Main Results

Hereafter, all limit relationships hold as $x\to \mathrm{\infty }$ unless stated otherwise. For two positive functions $a(x)$ and $b(x)$, we write $a(x)\lesssim b(x)$, if $\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}a(x)/b(x)\le \mathrm{1}$; write $a(x)\gtrsim b(x)$, if $\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}a(x)/b(x)\ge \mathrm{1}$ and write $a(x)~b(x)$, if $\mathrm{l}\mathrm{i}\mathrm{m}a(x)/b(x)=\mathrm{1}$. For two positive functions $a(x,t)$ and $b(x,t)$, we say that $a(x,t)\lesssim b(x,t)$ holds uniformly for $t\in \mathrm{\Delta }\ne \mathrm{\varnothing }$, If

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in \mathrm{\Delta }}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{a(x,t)}{b(x,t)}\le \mathrm{1}$;

say that $a(x,t)\gtrsim b(x,t)$ holds uniformly for $t\in \mathrm{\Delta }\ne \mathrm{\varnothing }$, if

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\underset{t\in \mathrm{\Delta }}{\mathrm{i}\mathrm{n}\mathrm{f}}\frac{a(x,t)}{b(x,t)}\ge \mathrm{1}$;

and say that $a(x,t)~b(x,t)$ holds uniformly for $t\in \mathrm{\Delta }\ne \mathrm{\varnothing }$, if $a(x,t)\gtrsim b(x,t)$ and $a(x,t)\lesssim b(x,t)$ hold uniformly for $t\in \mathrm{\Delta }\ne \mathrm{\varnothing }$. ${\mathrm{1}}_A$ is the indicator function of a set $A$.

In this paper, we will consider the claim sizes have heavy-tailed distributions. Some subclasses of heavy-tailed distribution class will be given. Say that a distribution $V$ on $(-\mathrm{\infty },\mathrm{\infty })$ is heavy-tailed if for any $\lambda >\mathrm{0}$,

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\lambda t}V(\mathrm{d}t)=\mathrm{\infty }.$

One of the important distribution classes of heavy-tailed distributions is the consistently-varying-tailed distribution class $\mathcal{C}$. By definition, a distribution $V$ on $(-\mathrm{\infty },\mathrm{\infty })$ belongs to the class $\mathcal{C}$, denoted by $V\in \mathcal{C}$, if

$\underset{y\nearrow \mathrm{1}}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\overline{V}(xy)}{\overline{V}(x)}=\mathrm{1},$

or equivalently,

$\underset{y\searrow \mathrm{1}}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}\frac{\overline{V}(xy)}{\overline{V}(x)}=\mathrm{1}.$

A related distribution class is the dominated varying tailed distribution class $\mathcal{D}$ . Say that a distribution $V$ on $(-\mathrm{\infty },\mathrm{\infty })$ belongs to the class $\mathcal{D}$ , denoted by $V\in \mathcal{D}$ , if for any fixed $\mathrm{0}<y<\mathrm{1}$,

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\overline{V}(xy)}{\overline{V}(x)}<\mathrm{\infty }.$

A distribution $V$ on $(-\mathrm{\infty },\mathrm{\infty })$ is said to be in the long-tailed distribution class $\mathrm{ℒ}$, if for any fixed $y>\mathrm{0}$,

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\overline{V}(x+y)}{\overline{V}(x)}=\mathrm{1}.$

An important subclass of the class $\mathrm{ℒ}$ is the subexponential distribution class $\mathcal{S}$. By definition, a distribution $V$ on $[\mathrm{0},\mathrm{\infty })$ is said to be subexponential if

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\overline{V\mathrm{*}V}(x)}{\overline{V}(x)}=\mathrm{2},$

where $V\mathrm{*}V$ denotes the $\mathrm{2}$-fold convolution of $V$. In the case that a distribution $V$ is on $(-\mathrm{\infty },\mathrm{\infty })$, we say that $V\in \mathcal{S}$ if the distribution $V(x){\mathrm{1}}_{\{x\ge \mathrm{0}\}}$ belongs to the class $\mathcal{S}$. It is well-known that these distribution classes have the following inclusions

$\mathcal{C}\subset \mathrm{ℒ}\bigcap \mathcal{D}\subset \mathcal{S}\subset \mathrm{ℒ}$

see, e.g., Embrechts et al ^[14]. Korshunov ^[15] introduced another subclass of the subexponential distribution class, which is the strongly subexponential distribution class ${\mathcal{S}}_{\mathrm{*}}$. Say that a distribution $V$ on $(-\mathrm{\infty },\mathrm{\infty })$ belongs to the class ${\mathcal{S}}_{\mathrm{*}}$, if ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\overline{V}(y)\mathrm{d}y<\mathrm{\infty }$ and the distribution $V_u$ defined by

$\overline{V_u}(x)=\{\begin{array}{c}\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{1},{\int }_x^{x+u}\overline{V}(y)\mathrm{d}y\right\},\begin{array}{cc}& x\ge \mathrm{0},\end{array}\\ \mathrm{1},\begin{array}{cccc}& & & \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \end{array}x<\mathrm{0},\end{array}\end{array}$

satisfies

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\overline{V_u\mathrm{*}{V}_u}(x)}{\overline{V_u}(x)}=\mathrm{2}$

uniformly for $u\in [\mathrm{1},\mathrm{\infty })$. Korshunov ^[15] pointed out that the Pareto distribution with parameter exceeding one, the lognormal distribution and the Weibull distribution with suitably chosen parameters belong to the class ${\mathcal{S}}_{\mathrm{*}}$ and the class ${\mathcal{S}}_{\mathrm{*}}$ almost coincides with the class of subexponential distributions with finite means. For the distributions with finite means the following relationships hold

$\mathrm{ℒ}\bigcap \mathcal{D}\subset {\mathcal{S}}_{\mathrm{*}}\subset \mathcal{S}$

see, e.g., Korshunov ^[15] and Kaas et al^[16].

This paper mainly considers the claim-number process $\left\{N(t),t\ge \mathrm{0}\right\}$ satisfying the LDP. We first present the following assumption.

Assumption A 1) $P(N(t)/t\in \cdot )$ satisfies the LDP with rate function $I(\cdot )$ such that $I(x)=\mathrm{0}$ if and only if $x=z$, where $z$ is a positive constant.

2) $I(\cdot )$ is increasing on $[z,\mathrm{\infty })$ and decreasing on $[\mathrm{0},z]$.

As noted in Remark 2.1 of Fu et al^[10], the linear Hawkes process defined in Section 1 of Bordenave et al^[13] satisfies Assumption A. One can see Lefevere et al^[17], Macci et al ^[18] and Jiang et al^[19] for some other counting processes satisfying the LDP.

The following is the main result of this section.

Theorem 1   Consider the risk model (1). Suppose that Assumption A holds. If $F\in {\mathcal{S}}_{\mathrm{*}}$ and $v≔c/z-E\left(X_{\mathrm{1}}\right)>\mathrm{0}$, then

$\psi (x,t)~\frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(y)\mathrm{d}y$(3)

holds uniformly for $t\in [f(x),\mathrm{\infty })$, where $f:[\mathrm{0},\mathrm{\infty })\to [\mathrm{0},\mathrm{\infty })$ is an infinitely increasing function.

Before giving the proof of Theorem 1, we first present a lemma, which follows from Lemmas 1 and 9 in Korshunov ^[15](see also Lemma 2.2 in Leipus and Šiaulys^[3]).

Lemma 1   Let $\{{\xi }_i,i\ge \mathrm{1}\}$ be i.i.d. random variables with common distribution $V$ and finite mean $E{\xi }_{\mathrm{1}}<\mathrm{0}$.

1) If $V\in \mathrm{ℒ}$ , then for sufficiently large $x$,

$P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=\mathrm{1}}^k}{\xi }_i>x\right)\ge \frac{\mathrm{1}-{\epsilon }_{\mathrm{1}}(x)}{\left|E{\xi }_{\mathrm{1}}\right|}{\int }_x^{x+n\left|E{\xi }_{\mathrm{1}}\right|}\overline{V}(u)\mathrm{d}u$

holds uniformly for integers $n\ge \mathrm{1}$;

2) If $V\in {\mathcal{S}}_{\mathrm{*}}$, then for sufficiently large $x$,

$P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=\mathrm{1}}^k}{\xi }_i>x\right)\le \frac{\mathrm{1}+{\epsilon }_{\mathrm{2}}(x)}{\left|E{\xi }_{\mathrm{1}}\right|}{\int }_x^{x+n\left|E{\xi }_{\mathrm{1}}\right|}\overline{V}(u)\mathrm{d}u$

holds uniformly for integer $n\ge \mathrm{1}$, where ${\epsilon }_{\mathrm{1}}(x)$ and ${\epsilon }_{\mathrm{2}}(x)$ are some positive vanishing functions as $x\to \mathrm{\infty }$.

In the following we prove Theorem 1.

Proof of Theorem 1   By Assumption A, for any fixed $w_{\mathrm{1}}<z$ and $w_{\mathrm{2}}>z$, there exist some constants ${\delta }_{\mathrm{1}}>\mathrm{0}$ and ${\delta }_{\mathrm{2}}>\mathrm{0}$ such that $I(w_{\mathrm{1}})-{\delta }_{\mathrm{1}}>\mathrm{0},$$I(w_{\mathrm{2}})-{\delta }_{\mathrm{2}}>\mathrm{0}$ and for sufficiently large $t$,

$P(N(t)/t\le w_{\mathrm{1}})\le {\mathrm{e}}^{-t(I(w_{\mathrm{1}})-{\delta }_{\mathrm{1}})}$(4)

and

$P(N(t)/t\ge w_{\mathrm{2}})\le {\mathrm{e}}^{-t(I(w_{\mathrm{2}})-{\delta }_{\mathrm{2}})}$(5)

where the facts $I(x)>\mathrm{0}$ for $x\ne z$ and $I(\cdot )$ is decreasing on $[\mathrm{0},z]$ and increasing on $[z,\mathrm{\infty })$ have been used.

Note that for all $x\ge \mathrm{0}$ and $t\ge \mathrm{0}$,

$\psi (x,t)=P\left(\underset{\mathrm{1}\le k\le N(t)}{\mathrm{s}\mathrm{u}\mathrm{p}}\left({\displaystyle \sum _{i=\mathrm{1}}^k}{X}_i-c{\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i\right)>x\right).$

For any infinitely increasing function $f(x)$, we will prove

$\psi (x,t)\gtrsim \frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(y)\mathrm{d}y$(6)

and

$\psi (x,t)\lesssim \frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(y)\mathrm{d}y$(7)

hold uniformly for $t\in [f(x),\mathrm{\infty })$, respectively.

Firstly, we show the asymptotic upper bound (7). For any $\epsilon >\mathrm{0},x\ge \mathrm{0}$ and $t>\mathrm{0}$, we have

$\begin{array}{l}\psi (x,t)=P\left(\underset{\mathrm{1}\le k\le N(t)}{\mathrm{s}\mathrm{u}\mathrm{p}}\left({\displaystyle \sum _{i=\mathrm{1}}^k}{X}_i-c{\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i\right)>x,N(t)\le (\mathrm{1}+\epsilon )zt\right)\\ +P\left(\underset{\mathrm{1}\le k\le N(t)}{\mathrm{s}\mathrm{u}\mathrm{p}}\left({\displaystyle \sum _{i=\mathrm{1}}^k}{X}_i-c{\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i\right)>x,N(t)>(\mathrm{1}+\epsilon )zt\right)\\ =:{\psi }_{\mathrm{1}}(x,t)+{\psi }_{\mathrm{2}}(x,t).\end{array}$

For any $\delta \in (\mathrm{0},v/c)$, let

$A=\underset{\mathrm{1}\le k\le (\mathrm{1}+\epsilon )zt}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(X_i-c\left(\frac{\mathrm{1}}{z}-\delta \right)\right),$

$B=c\underset{k\ge \mathrm{1}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(\left(\frac{\mathrm{1}}{z}-\delta \right)-{\theta }_i\right)\mathrm{a}\mathrm{n}\mathrm{d}{B}^{+}≔\mathrm{m}\mathrm{a}\mathrm{x}\{B,\mathrm{0}\}.$

It follows from the conditions of the risk model that $A$ and $B^{+}$ are independent.

We first estimate ${\psi }_{\mathrm{1}}(x,t)$. Let $C≔E\left(X_{\mathrm{1}}-c\left(\mathrm{1}/z-\delta \right)\right)=c\delta -v<\mathrm{0}$. Therefore, for all $x\ge \mathrm{0},$$y\in (\mathrm{0},x/\mathrm{2}]$ and $t>\mathrm{0},$

$\begin{array}{l}{\psi }_{\mathrm{1}}(x,t)\le P(\underset{\mathrm{1}\le k\le (\mathrm{1}+\epsilon )zt}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(X_i-c\left(\frac{\mathrm{1}}{z}-\delta \right)\right)\\ +c\underset{k\ge \mathrm{1}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(\left(\frac{\mathrm{1}}{z}-\delta \right)-{\theta }_i\right)>x)\\ \le P\left(A+B^{+}>x\right)\\ \le {\int }_{\mathrm{0}}^{x-y}P\left(A>x-u\right)P\left(B^{+}\in \mathrm{d}u\right)+P\left(B^{+}>x-y\right)\\ =:{\psi }_{\mathrm{11}}(x,t)+{\psi }_{\mathrm{12}}(x,t)\end{array}$

Using the line of the proof of Proposition 2.1 of Leipus and Šiaulys ^[3], we know that for all $x\ge \mathrm{0},y\in (\mathrm{0},x/\mathrm{2}]$ and $t>\mathrm{0}$,

$\begin{array}{l}{\int }_{\mathrm{0}}^{x-y}P\left(A>x-u\right)P\left(B^{+}\in \mathrm{d}u\right)\\ \le \frac{(\mathrm{1}+\alpha (y))}{\left|C\right|}{\int }_{\mathrm{0}}^{x-y}\left({\int }_{x-u}^{x-u+vzt(\mathrm{1}+\epsilon )}\overline{F}(v)\mathrm{d}v\right)P\left(B^{+}\in \mathrm{d}u\right),\end{array}$

where $\alpha (\cdot )$ is a positive function satisfying $\underset{y\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\alpha (y)=\mathrm{0}.$ Let $J$ denote the integral of the right side in the above inequality and $G_{B^{+}}$ be the distribution of the random variable $B^{+}$. By Fubini's theorem, for all $x\ge \mathrm{0},y\in (\mathrm{0},x/\mathrm{2}]$ and $t>\mathrm{0}$,

$J={\int }_{\mathrm{0}}^{x-y}\left({\int }_x^{x+vzt(\mathrm{1}+\epsilon )}\overline{F}(w-u)\mathrm{d}w\right)G_{B^{+}}(du)\le {\int }_x^{x+vzt(\mathrm{1}+\epsilon )}\overline{F\mathrm{*}{G}_{B^{+}}}(w)\mathrm{d}w.$

Let $w_{\mathrm{2}}={(\mathrm{1}/z-\delta )}^{-\mathrm{1}}$ in (5). Since $\mathrm{0}<\delta <v/c<\mathrm{1}/z$, it knows that $w_{\mathrm{2}}>z$. By (5) there exists a constant ${\delta }_{\mathrm{2}}>\mathrm{0}$ such that for sufficiently large $x$,

$P\left(B>x\right)\le {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}P\left({\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i<k\left(\frac{\mathrm{1}}{z}-\delta \right)-\frac{x}{c}\right)$

$\le \sum _{k\ge \frac{xz/c}{\mathrm{1}-z\delta }}P\left({\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i<k\left(\frac{\mathrm{1}}{z}-\delta \right)\right)$

$\le \sum _{k\ge \frac{xz/c}{\mathrm{1}-z\delta }}P\left(N\left(k\left(\frac{\mathrm{1}}{z}-\delta \right)\right)\ge k\right)$

$\le \sum _{k\ge \frac{xz/c}{\mathrm{1}-z\delta }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-k\left(\frac{\mathrm{1}}{z}-\delta \right)\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)\right)$

$\le \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{x}{c}\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)\right)}{\mathrm{1}-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\left(\frac{\mathrm{1}}{z}-\delta \right)\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)\right)}$

$=:d_{\mathrm{1}}\mathrm{e}\mathrm{x}\mathrm{p}(-d_{\mathrm{2}}x)$(8)

where

$d_{\mathrm{1}}={\left(\mathrm{1}-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\left(\frac{\mathrm{1}}{z}-\delta \right)\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)\right)\right)}^{-\mathrm{1}}>\mathrm{0}$

and

$d_{\mathrm{2}}=c^{-\mathrm{1}}\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)>\mathrm{0}.$

Thus for $x>\mathrm{0},$

${\overline{G}}_{B^{+}}(x)=P\left(B^{+}>x\right)=P\left(B>x\right)\le d_{\mathrm{1}}\mathrm{e}\mathrm{x}\mathrm{p}(-d_{\mathrm{2}}x).$

Since $F\in {\mathcal{S}}_{\mathrm{*}}\subset \mathcal{S}$, it holds that ${\overline{G}}_{B^{+}}(x)=o\left(\overline{F}(x)\right).$ By Corollary 3.18 of Foss et al^[20],

$\overline{F\mathrm{*}{G}_{B^{+}}}(x)~\overline{F}(x).$

Consequently, there exists a positive function $\beta (x)\to \mathrm{0}$ such that for sufficiently large $x$,

$J\le (\mathrm{1}+\beta (x)){\int }_x^{x+vzt(\mathrm{1}+\epsilon )}\overline{F}(u)\mathrm{d}u$

$=(\mathrm{1}+\beta (x)){\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u\left(\mathrm{1}+\frac{{\int }_{x+vzt}^{x+vzt(\mathrm{1}+\epsilon )}\overline{F}(u)\mathrm{d}u}{{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\right)$

$\le (\mathrm{1}+\beta (x))(\mathrm{1}+\epsilon ){\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u$(9)

So, for all $t>\mathrm{0},y\in (\mathrm{0},x/\mathrm{2}]$ and sufficiently large $x$,

${\int }_{\mathrm{0}}^{x-y}P\left(A>x-u\right)P\left(B^{+}\in \mathrm{d}u\right)$

$\le \frac{(\mathrm{1}+\alpha (y))(\mathrm{1}+\beta (x))(\mathrm{1}+\epsilon )}{\left|C\right|}{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u$(10)

By (10), it holds for all $t>\mathrm{0},y\in (\mathrm{0},x/\mathrm{2}]$ and sufficiently large $x$ that

${\psi }_{\mathrm{11}}(x,t)\le \frac{(\mathrm{1}+\alpha (y))(\mathrm{1}+\beta (x))(\mathrm{1}+\epsilon )}{\left|C\right|}{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u,$

which shows that

$\underset{\epsilon \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{\delta \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\psi }_{\mathrm{11}}(x,t)}{\frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\le \mathrm{1}$(11)

In the following, we deal with ${\psi }_{\mathrm{12}}(x,t)$. Using (8), for sufficiently large $x$, we have

$P\left(B^{+}>x-y\right)\le \sum _{k\ge \frac{(x-y)z}{c(\mathrm{1}-z\delta )}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-k\left(\frac{\mathrm{1}}{z}-\delta \right)\left(I\left({\left(\frac{\mathrm{1}}{z}-\delta \right)}^{-\mathrm{1}}\right)-{\delta }_{\mathrm{2}}\right)\right)$

$\le d_{\mathrm{1}}\mathrm{e}\mathrm{x}\mathrm{p}(-d_{\mathrm{2}}(x-y)).$

Since $t\ge f(x)$ and $f(x)\uparrow \mathrm{\infty }$ as $x\to \mathrm{\infty }$, for sufficiently large $x$, it holds that $t\ge \mathrm{1}/vz$. Therefore, by $F\in {\mathcal{S}}_{\mathrm{*}}\subset \mathcal{S}\subset \mathrm{ℒ}$, for all $t\ge f(x)$ and $y\in (\mathrm{0},x/\mathrm{2}]$, it holds that

$\frac{P\left(B^{+}>x-y\right)}{{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\le \frac{P\left(B^{+}>\frac{x}{\mathrm{2}}\right)}{{\int }_x^{x+\mathrm{1}}\overline{F}(u)\mathrm{d}u}\le \frac{d_{\mathrm{1}}{\mathrm{e}}^{-d_{\mathrm{2}}x/\mathrm{2}}}{\overline{F}(x+\mathrm{1})}\to \mathrm{0}$(12)

Combining with (12) yields that

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\psi }_{\mathrm{12}}(x,t)}{\frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(y)\mathrm{d}y}=\mathrm{0}$(13)

By (11) and (13), we get

$\underset{\epsilon \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\psi }_{\mathrm{1}}(x,t)}{\frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\le \mathrm{1}$(14)

Next, we will estimate the asymptotic upper bound of ${\psi }_{\mathrm{2}}(x,t)$. We first deal with the process $\{N(t),t\ge \mathrm{0}\}$. By (5), for $\widetilde{w_{\mathrm{2}}}=(\mathrm{1}+\epsilon )z>z$, there exists $\widetilde{{\delta }_{\mathrm{2}}}>\mathrm{0}$ such that $I((\mathrm{1}+\epsilon )z)-\widetilde{{\delta }_{\mathrm{2}}}>\mathrm{0}$ and for all $t\in [f(x),\mathrm{\infty }),\gamma >\mathrm{0}$ and sufficiently large $x$,

$\sum _{n>(\mathrm{1}+\epsilon )zt}{(\mathrm{1}+\gamma )}^{n}P(N(t)\ge n)\le \sum _{n>(\mathrm{1}+\epsilon )zt}{(\mathrm{1}+\gamma )}^{n}P\left(N\left(\frac{n}{(\mathrm{1}+\epsilon )z}\right)\ge n\right)$

$\le \sum _{n>(\mathrm{1}+\epsilon )zt}{(\mathrm{1}+\gamma )}^n\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{n}{(\mathrm{1}+\epsilon )z}\left(I\left((\mathrm{1}+\epsilon )z\right)-\widetilde{{\delta }_{\mathrm{2}}}\right)\right)$(15)

It holds for all $x\ge \mathrm{0}$ and $t\ge \mathrm{0}$ that

${\psi }_{\mathrm{2}}(x,t)\le \sum _{n>(\mathrm{1}+\epsilon )zt}P\left(\underset{\mathrm{1}\le k\le n}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}{X}_i>x,N(t)=n\right)$

$=\sum _{n>(\mathrm{1}+\epsilon )zt}\overline{F^{\mathrm{*}n}}(x)P\left(N(t)=n\right)$(16)

Since $F\in {\mathcal{S}}_{\mathrm{*}}\subset \mathcal{S}$, by Kesten's bound, for $\mathrm{0}<\gamma <\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\left(I\left((\mathrm{1}+\epsilon )z\right)-\widetilde{{\delta }_{\mathrm{2}}}\right)}{\left((\mathrm{1}+\epsilon )z\right)}\right)-\mathrm{1}$, there exists $l=l(\gamma )>\mathrm{0}$ such that for any $x\ge \mathrm{0}$ and $n\ge \mathrm{1}$,

$\overline{F^{\mathrm{*}n}}(x)\le l{(\mathrm{1}+\gamma )}^n\overline{F}(x)$(17)

Therefore, by $F\in {\mathcal{S}}_{\mathrm{*}}\subset \mathrm{ℒ}$ and (15)-(17),

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\psi }_{\mathrm{2}}(x,t)}{{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\le \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\overline{F}(x)}{{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}$

$\cdot \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}l\sum _{n>(\mathrm{1}+\epsilon )zt}{(\mathrm{1}+\gamma )}^n\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{n}{(\mathrm{1}+\epsilon )z}\left(I\left((\mathrm{1}+\epsilon )z\right)-\widetilde{{\delta }_{\mathrm{2}}}\right)\right)$

$\begin{array}{l}\le \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\overline{F}(x)}{\overline{F}(x+\mathrm{1})}\\ \cdot \underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}l\sum _{n>(\mathrm{1}+\epsilon )zt}{(\mathrm{1}+\gamma )}^n\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{n}{(\mathrm{1}+\epsilon )z}\left(I\left((\mathrm{1}+\epsilon )z\right)-\widetilde{{\delta }_{\mathrm{2}}}\right)\right)\end{array}$

$=\mathrm{0}$(18)

Thus, by (14) and (18),

$\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}}\underset{t\in [f(x),\mathrm{\infty })}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{\psi (x,t)}{\frac{\mathrm{1}}{v}{\int }_x^{x+vzt}\overline{F}(u)\mathrm{d}u}\le \mathrm{1}.$

This completes the proof of (7).

Next we prove the asymptotic lower bound (6). For any $\epsilon >\mathrm{0}$, by (4), for $w_{\mathrm{1}}=\frac{z}{\mathrm{1}+z\epsilon }<z$, there exists ${\delta }_{\mathrm{1}}>\mathrm{0}$ such that $I\left(\frac{z}{\mathrm{1}+z\epsilon }\right)-{\delta }_{\mathrm{1}}>\mathrm{0}$ and for sufficiently large $M$,

$P\left(c\underset{k\ge \mathrm{1}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left({\theta }_i-(\mathrm{1}/z+\epsilon )\right)\ge M\right)$

$\le {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}P\left({\displaystyle \sum _{i=\mathrm{1}}^k}{\theta }_i\ge \frac{M+kc(\mathrm{1}/z+\epsilon )}{c}\right)$

$\le {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}P\left(N\left(\frac{M+kc(\mathrm{1}/z+\epsilon )}{c}\right)\le k\right)$

$\le {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}P\left(N\left(\frac{M+kc(\mathrm{1}/z+\epsilon )}{c}\right)\le \frac{M+kc(\mathrm{1}/z+\epsilon )}{c}\cdot \frac{z}{\mathrm{1}+z\epsilon }\right)$

$\le {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{M+kc(\mathrm{1}/z+\epsilon )}{c}\left(I\left(z/(\mathrm{1}+z\epsilon )\right)-{\delta }_{\mathrm{1}}\right)\right)$

$=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{M}{c}\left(I\left(z/(\mathrm{1}+z\epsilon )\right)-{\delta }_{\mathrm{1}}\right)\right)$

$\cdot {\displaystyle \sum _{k=\mathrm{1}}^{\mathrm{\infty }}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-k(\mathrm{1}/z+\epsilon )\left(I\left(z/(\mathrm{1}+z\epsilon )\right)-{\delta }_{\mathrm{1}}\right)\right)$

$\to \mathrm{0}\hspace{1em}\mathrm{a}\mathrm{s}\hspace{1em}M\to \mathrm{\infty }$(19)

Thus, by (19), for any $\mathrm{0}<\epsilon <\mathrm{1}$,

$\underset{M\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}P\left(c\underset{k\ge \mathrm{1}}{\mathrm{i}\mathrm{n}\mathrm{f}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(\frac{\mathrm{1}}{z}+\epsilon -{\theta }_i\right)>-M\right)$

$=\underset{M\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}P\left(c\underset{k\ge \mathrm{1}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left({\theta }_i-\left(\frac{\mathrm{1}}{z}+\epsilon \right)\right)<M\right)=\mathrm{1}$(20)

For the above $\epsilon >\mathrm{0}$, let $\tilde{v}≔c(\mathrm{1}/z+\epsilon )-E{X}_{\mathrm{1}},$ then $\tilde{v}>\mathrm{0}$ and $\tilde{v}\to v$ as $\epsilon \to \mathrm{0}$. Thus, for the above $\epsilon >\mathrm{0}$ and $M>\mathrm{0}$, and for all $t\in [f(x),\mathrm{\infty })$, by Lemma 1,

$\psi (x,t)$

$\ge P\left(\underset{\mathrm{1}\le k\le N(t)}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(X_i-c\left(\frac{\mathrm{1}}{z}+\epsilon \right)\right)+c\underset{k\ge \mathrm{1}}{\mathrm{i}\mathrm{n}\mathrm{f}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(\frac{\mathrm{1}}{z}+\epsilon -{\theta }_i\right)>x\right)$

$\ge P(\underset{\mathrm{1}\le k\le N(t)}{\mathrm{s}\mathrm{u}\mathrm{p}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(X_i-c\left(\frac{\mathrm{1}}{z}+\epsilon \right)\right)>x+M,$

$c\underset{k\ge \mathrm{1}}{\mathrm{i}\mathrm{n}\mathrm{f}}{\displaystyle \sum _{i=\mathrm{1}}^k}\left(\frac{\mathrm{1}}{z}+\epsilon -{\theta }_i\right)>-M)$