1 Introduction
The classical risk model and the renewal risk model are two models that are traditionally used to describe the nonlife insurance business. The classical risk model was introduced by Lundberg and Cramér about a century ago (see [8, 14, 15] for the source papers and [18] for the historical environment). In this risk model, it is assumed that interarrival times are identically distributed, exponential, and independent random variables. In 1957, the Danish mathematician E. Sparre Andersen proposed the renewal risk model to describe the surplus process of the insurance company. In the renewal risk model, the claim sizes and the interarrival times are independent, identically distributed, nonnegative random variables (see [2] for the source paper and [22] for additional details). In this paper, we assume that interoccurrence times and claim sizes are nonnegative random variables (r.v.s) that are not necessarily identically distributed. We call such a model the inhomogeneous model and present its exact definition. It is evident that the inhomogeneous renewal risk model reflects better the real insurance activities in comparison with the classical risk model or with the renewal (homogeneous) risk model.
Definition 1.
We say that the insurer’s surplus $U(t)$ varies according to the inhomogeneous renewal risk model if
for all $t\geqslant 0$. Here:
-
• $x\geqslant 0$ is the initial reserve;
-
• claim sizes $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ form a sequence of independent (not necessarily identically distributed) nonnegative r.v.s;
-
• $c>0$ is the constant premium rate;
-
• $\varTheta (t)={\sum _{n=1}^{\infty }}\mathbb{1}_{\{T_{n}\leqslant t\}}=\sup \{n\geqslant 0:T_{n}\leqslant t\}$ is the number of claims in the interval $[0,t]$, where $T_{0}=0$, $T_{n}=\theta _{1}+\theta _{2}+\cdots +\theta _{n}$, $n\geqslant 1$, and the interarrival times $\{\theta _{1},\theta _{2},\dots \}$ are independent (not necessarily identically distributed), nonnegative, and nondegenerate at zero r.v.s;
-
• the sequences $\{Z_{1},Z_{2},\dots \}$ and $\{\theta _{1},\theta _{2},\dots \}$ are mutually independent.
A typical path of the surplus process of an insurance company is shown in Fig. 1.
If all claim sizes $\{Z_{1},Z_{2},\dots \}$ and all interarrival times $\{\theta _{1},\theta _{2},\dots \}$ are identically distributed, then the inhomogeneous renewal risk model becomes the homogeneous renewal risk model.
The time of ruin and the ruin probability are the main critical characteristics of any risk model. Let $\mathcal{B}$ denote the event of ruin. We suppose that
\[ \mathcal{B}=\bigcup \limits_{t\geqslant 0}\big\{\omega :U(\omega ,t)<0\big\}=\bigcup \limits_{t\geqslant 0}\Bigg\{\omega :x+ct-{\sum \limits_{i=1}^{\varTheta (t)}}Z_{i}<0\Bigg\},\]
that is, that ruin occurs if at some time $t\geqslant 0$ the surplus of the insurance company becomes negative or, in other words, the insurer becomes unable to pay all the claims. The first time τ when the surplus drops to a level less than zero is called the time of ruin, that is, τ is the extended r.v. for which
The ruin probability ψ is defined by the equality
Usually, we suppose that the main parameter of the ruin probability is the initial reserve x, though actually the ruin probability, together with time of ruin, depends on all components of the renewal risk model.
All trajectories of the process $U(t)$ are increasing functions between times $T_{n}$ and $T_{n+1}$ for all $n=0,1,2,\dots $ . Therefore, the random variables $U(\theta _{1}+\theta _{2}+\cdots +\theta _{n})$, $n\geqslant 1$, are the local minimums of the trajectories. Consequently, we can express the ruin probability in the following way (for details, see [9] or [16]):
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \psi (x)& \displaystyle =\mathbb{P}\Big(\underset{n\in \mathbb{N}}{\inf }U(\theta _{1}+\theta _{2}+\cdots +\theta _{n})<0\Big)\\{} & \displaystyle =\mathbb{P}\Bigg(\underset{n\in \mathbb{N}}{\inf }\Bigg\{x+c(\theta _{1}+\theta _{2}+\cdots +\theta _{n})-{\sum \limits_{i=1}^{\varTheta (\theta _{1}+\cdots +\theta _{n})}}\hspace{-0.1667em}Z_{i}\Bigg\}<0\Bigg)\\{} & \displaystyle =\mathbb{P}\Bigg(\underset{n\in \mathbb{N}}{\inf }\Bigg\{x-{\sum \limits_{i=1}^{n}}(Z_{i}-c\theta _{i})\Bigg\}<0\Bigg)\\{} & \displaystyle =\mathbb{P}\Bigg(\underset{n\in \mathbb{N}}{\sup }\Bigg\{{\sum \limits_{i=1}^{n}}(Z_{i}-c\theta _{i})\Bigg\}>x\Bigg).\end{array}\]
Further, in this paper, we restrict our study to the so-called Lundberg-type inequality. An exponential bound for the ruin probability is usually called a Lundberg-type inequality. We further give the well-known exponential bound for $\psi (x)$ in homogeneous renewal risk model (see, for instance, Chapters “Lundberg Inequality for Ruin Probability”, “Collective Risk Theory”, “Adjustment Coefficient,” or “Cramer–Lundberg Asymptotics” in [21]).
Theorem 1.
Let the net profit condition $\mathbb{E}Z_{1}-c\mathbb{E}\theta _{1}<0$ hold, and let $\mathbb{E}{\mathrm{e}}^{hZ_{1}}<\infty $ for some $h>0$ in the homogeneous renewal risk model. Then, there is a number $H>0$ such that
for all $x\geqslant 0$. If $\mathbb{E}{\mathrm{e}}^{R(Z_{1}-c\theta _{1})}=1$ for some positive R, then we can take $H=R$ in (1).
There exist a lot of different proofs of this theorem. The main ways to prove inequality (1) are described in Chapter “Lundberg Inequality for Ruin Probability” of [21]. Details of some existing proofs were given, for instance, by Asmussen and Albrecher [3], Embrechts, Klüppelberg, and Mikosch [9], Embrechts and Veraverbeke [10], Gerber [11], and Mikosch [16]. We note only that the bound (1) can be proved using the exponential tail bound of Sgibnev [19] and the inequality $\psi (0)<1$.
The following theorem is the main statement of the paper.
Theorem 2.
Let the claim sizes $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ and the interarrival times $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ form an inhomogeneous renewal risk model described in Definition 1. Further, let the following three conditions be satisfied:
\[\begin{array}{r@{\hskip0pt}l}\displaystyle (\mathcal{C}1)\hspace{2.84544pt}& \displaystyle \underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\gamma Z_{i}}<\infty \hspace{1em}\textit{for some}\hspace{2.5pt}\gamma >0,\\{} \displaystyle (\mathcal{C}2)\hspace{2.84544pt}& \displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(\theta _{i}\mathbb{1}_{\{\theta _{i}>u\}}\big)=0,\\{} \displaystyle (\mathcal{C}3)\hspace{2.84544pt}& \displaystyle \underset{n\to \infty }{\limsup }\frac{1}{n}{\sum \limits_{i=1}^{n}}(\hspace{0.1667em}\mathbb{E}Z_{i}-c\mathbb{E}\theta _{i})<0.\end{array}\]
Then, there are constants $c_{1}>0$ and $c_{2}\geqslant 0$ such that $\psi (x)\leqslant {\mathrm{e}}^{-c_{1}x}$ for all $x\geqslant c_{2}$.
The inhomogeneous renewal risk model differs from the homogeneous one because the independence and/or homogeneous distribution of sequences of random variables $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ and/or $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ are no longer required. The changes depend on how the inhomogeneity in a particular model is understood. In Definition 1, we have chosen one of two possible directions used in numerous articles that deal with inhomogeneous renewal risk models. This is due to the fact that an inhomogeneity can be considered as the possibility to have either differently distributed or dependent r.v.s in the sequences.
The possibility to have differently distributed r.v.s was considered, for instance, in [5, 6, 12, 17]. In the first three works, the discrete-time inhomogeneous risk model was considered. In such a model, the interarrival times are fixed, and the claims $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ are independent, not necessary identically distributed, integer valued r.v.s. In [17], the authors consider the model where the interarrival times are identically distributed and have a particular distribution, whereas the claims are differently distributed with distributions belonging to a particular class. Bernackaitė and Šiaulys [4] deal with an inhomogeneous renewal risk model where the r.v.s $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ are not necessarily identically distributed, but the claim sizes $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ have a common distribution function. In this article, we consider a more general renewal risk model. In the main theorem, we assume that not only r.v.s $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ are not necessarily identically distributed, but also the same holds for the sequence of claim sizes $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$.
There is another approach to inhomogeneous renewal risk models, which implies the possibility to have dependence in sequences and mainly found in works by Chinese researchers. In this kind of models, the sequences $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ and $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ consist of identically distributed r.v.s, but there may be some kind of dependence between them. Results for such models can be found, for instance, in [7] and [23]. Another interpretation of dependence is also possible, where r.v.s in both sequences $\{Z_{1},Z_{2},\dots \hspace{0.1667em}\}$ and $\{\theta _{1},\theta _{2},\dots \hspace{0.1667em}\}$ still remain independent. Instead of that, the mutual independence of these two sequences is no longer required. The idea of this kind of dependence belongs to Albrecher and Teugels [1], and this encouraged Li, Tang, and Wu [13] to study renewal risk models having this dependence structure.
2 Auxiliary lemma
In order to prove the main theorem, we need an auxiliary lemma. In Lemma 1, the conditions for r.v.s $\eta _{1}$, $\eta _{2}$, $\eta _{3},\dots $ are taken from articles by Smith [20] and Bernackaitė and Šiaulys [4].
Lemma 1.
Let $\eta _{1}$, $\eta _{2}$, $\eta _{3},\dots $ be independent r.v.s such that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big(\mathcal{C}{1}^{\ast }\big)\hspace{2.84544pt}& \displaystyle \underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}}<\infty \hspace{1em}\textit{for some}\hspace{2.5pt}\delta >0,\\{} \displaystyle \big(\mathcal{C}{2}^{\ast }\big)\hspace{2.84544pt}& \displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)=0,\\{} \displaystyle \big(\mathcal{C}{3}^{\ast }\big)\hspace{2.84544pt}& \displaystyle \underset{n\to \infty }{\limsup }\frac{1}{n}{\sum \limits_{i=1}^{n}}\mathbb{E}\eta _{i}<0.\end{array}\]
Then, there are constants $c_{3}>0$ and $c_{4}>0$ such that
\[ \mathbb{P}\Bigg(\underset{k\geqslant 1}{\sup }{\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\leqslant c_{3}{\mathrm{e}}^{-c_{4}x}\]
for all $x\geqslant 0$.
Proof.
First, we observe that, for all $x\geqslant 0$,
(2)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{P}\Bigg(\underset{k\geqslant 1}{\sup }{\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)& \displaystyle =\mathbb{P}\Bigg({\bigcup \limits_{k=1}^{\infty }}\Bigg\{{\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg\}\Bigg)\\{} & \displaystyle \leqslant {\sum \limits_{k=1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg).\end{array}\]By Chebyshev’s inequality, for all $x\geqslant 0$, $0<y\leqslant \delta $, and $k\in \mathbb{N}$, we have
(3)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)& \displaystyle =\mathbb{P}\big({\mathrm{e}}^{y{\textstyle\sum _{i=1}^{k}}\eta _{i}}>{\mathrm{e}}^{yx}\big)\\{} & \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{y\eta _{i}}.\end{array}\]Moreover, for all $i\in \mathbb{N}$, $0<y\leqslant \delta $, and $u>0$, we have
and
(4)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}{\mathrm{e}}^{y\eta _{i}}=1+y\mathbb{E}\eta _{i}+\mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\end{array}\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\\{} & \displaystyle \hspace{1em}=\mathbb{E}\big(\big({\mathrm{e}}^{y\eta _{i}}-1\big)\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)-y\mathbb{E}(\eta _{i}\mathbb{1}_{\{\eta _{i}\leqslant -u\}})\\{} & \displaystyle \hspace{2em}+\mathbb{E}\big(\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\mathbb{1}_{\{-u<\eta _{i}\leqslant 0\}}\big)+\mathbb{E}\big(\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\mathbb{1}_{\{\eta _{i}>0\}}\big).\end{array}\]
In order to evaluate the absolute value of the remainder term in (4), we need the following inequalities:
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big|{\mathrm{e}}^{v}-1\big|\leqslant |v|,& \displaystyle \hspace{1em}v\leqslant 0,\\{} \displaystyle \big|{\mathrm{e}}^{v}-v-1\big|\leqslant \frac{{v}^{2}}{2},& \displaystyle \hspace{1em}v\leqslant 0,\\{} \displaystyle \big|{\mathrm{e}}^{v}-v-1\big|\leqslant \frac{{v}^{2}}{2}{\mathrm{e}}^{v},& \displaystyle \hspace{1em}v\geqslant 0.\end{array}\]
Using them, we get
where $i\in \mathbb{N}$, $0<y\leqslant \delta $, and $u>0$.
(5)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \big|\mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\big|\\{} & \displaystyle \hspace{1em}\leqslant 2y\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)+\frac{{y}^{2}}{2}\mathbb{E}\big({\eta _{i}^{2}}\mathbb{1}_{\{-u<\eta _{i}\leqslant 0\}}\big)+\frac{{y}^{2}}{2}\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{y\eta _{i}}\mathbb{1}_{\{\eta _{i}>0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant 2y\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)+\frac{{y}^{2}{u}^{2}}{2}+\frac{{y}^{2}}{2}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{y\eta _{i}}\mathbb{1}_{\{\eta _{i}>0\}}\big),\end{array}\]Since
we have
for all $v\geqslant c_{5}$, where $c_{5}=c_{5}(\delta )>0$.
Therefore,
(6)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{\delta \eta _{i}/2}\mathbb{1}_{\{\eta _{i}>0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{\delta \eta _{i}/2}\mathbb{1}_{\{0<\eta _{i}\leqslant c_{5}\}}\big)+\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{\delta \eta _{i}/2}\mathbb{1}_{\{\eta _{i}>c_{5}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \big({c_{5}^{2}}+1\big)\underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}}<\infty .\end{array}\]Choosing $u=\frac{1}{\sqrt[4]{y}}$ in (5) and using (6), we get
where $i\in \mathbb{N}$, $y\in (0,\delta /2]$, $c_{5}=c_{5}(\delta )$, and
(7)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \big|\mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\big|\\{} & \displaystyle \hspace{1em}\leqslant 2y\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\Big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -\frac{1}{\sqrt[4]{y}}\}}\Big)+\frac{{y}^{\frac{3}{2}}}{2}+\frac{{y}^{2}}{2}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{y\eta _{i}}\mathbb{1}_{\{\eta _{i}>0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant y\bigg(2\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -\frac{1}{\sqrt[4]{y}}\}}\big)+\frac{{y}^{\frac{1}{2}}}{2}+\frac{y}{2}\big({c_{5}^{2}}+1\big)\underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}}\bigg)\\{} & \displaystyle \hspace{1em}=:y\alpha (y),\end{array}\]Conditions ($\mathcal{C}{1}^{\ast }$) and ($\mathcal{C}{2}^{\ast }$) imply that $\alpha (y)\downarrow 0$ as $y\to 0$.
For an arbitrary positive v, we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}<0\}}\big)& \displaystyle =\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{-v<\eta _{i}<0\}}+|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -v\}}\big)\\{} & \displaystyle \leqslant v+\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -v\}}\big).\end{array}\]
So, condition ($\mathcal{C}{2}^{\ast }$) implies that
Denote
If $y\in (0,\widehat{y}\hspace{0.1667em}]$, then
\[\begin{array}{r@{\hskip0pt}l}\displaystyle y\big(\mathbb{E}\eta _{i}+\alpha (y)\big)& \displaystyle >y\mathbb{E}\eta _{i}\\{} & \displaystyle =y\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}\geqslant 0\}}+\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant y\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant \widehat{y}\underset{i\in \mathbb{N}}{\inf }\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle =-\widehat{y}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant -1/2\end{array}\]
for all $i\in \mathbb{N}$.Therefore, (3), (4), (7), and the well-known inequality
imply that
where $k\in \mathbb{N}$, $x\geqslant 0$, and $y\in (0,\widehat{y}\hspace{0.1667em}]$.
(9)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)& \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\big(1+y\mathbb{E}\eta _{i}+\mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\big)\\{} & \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\big(1+y\big(\mathbb{E}\eta _{i}+\alpha (y)\big)\big)\\{} & \displaystyle =\exp \Bigg\{-yx+{\sum \limits_{i=1}^{k}}\ln \big(1+y\big(\mathbb{E}\eta _{i}+\alpha (y)\big)\big)\Bigg\}\\{} & \displaystyle \leqslant \exp \Bigg\{-yx+y{\sum \limits_{i=1}^{k}}\mathbb{E}\eta _{i}+yk\alpha (y)\Bigg\},\end{array}\]By estimate (8) and condition ($\mathcal{C}{3}^{\ast }$) we can suppose that
for some positive constant $c_{6}$. Then we have
for $k\geqslant M+1$ with some $M\geqslant 1$. Moreover, there exists ${y}^{\ast }\in (0,\widehat{y}\hspace{0.1667em}]$ such that $\alpha ({y}^{\ast })\leqslant c_{6}/4$ since $\alpha (y)\downarrow 0$ as $y\to 0$.
Using results from (2), (3), and (9), we derive
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mathbb{P}\Bigg(\underset{k\geqslant 1}{\sup }{\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)+{\sum \limits_{k=M+1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}{\mathrm{e}}^{-{y}^{\ast }x}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=M+1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}{\mathrm{e}}^{-{y}^{\ast }x}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=M+1}^{\infty }}{\mathrm{e}}^{-{y}^{\ast }x+{y}^{\ast }{\textstyle\sum _{i=1}^{k}}\mathbb{E}\eta _{i}+{y}^{\ast }k\alpha ({y}^{\ast })}\\{} & \displaystyle \hspace{1em}\leqslant {\mathrm{e}}^{-{y}^{\ast }x}\Bigg({\sum \limits_{k=1}^{M}}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=0}^{\infty }}{\mathrm{e}}^{-k{y}^{\ast }c_{6}/4}\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\mathrm{e}}^{-{y}^{\ast }x}\Bigg({\sum \limits_{k=1}^{M}}{\prod \limits_{i=1}^{k}}\varDelta +\frac{1}{1-{\mathrm{e}}^{-{y}^{\ast }c_{6}/4}}\Bigg)\\{} & \displaystyle \hspace{1em}={\mathrm{e}}^{-{y}^{\ast }x}\bigg(\frac{\varDelta ({\varDelta }^{M}-1)}{\varDelta -1}+\frac{{\mathrm{e}}^{{y}^{\ast }c_{6}/4}}{{\mathrm{e}}^{{y}^{\ast }c_{6}/4}-1}\bigg)=:c_{3}{\mathrm{e}}^{-c_{4}x},\end{array}\]
where
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle x\geqslant 0,\\{} & \displaystyle \varDelta =1+\underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}},\\{} & \displaystyle c_{3}=\frac{\varDelta ({\varDelta }^{M}-1)}{\varDelta -1}+\frac{{e}^{{y}^{\ast }c_{6}/4}}{{e}^{{y}^{\ast }c_{6}/4}-1},\\{} & \displaystyle c_{4}={y}^{\ast }\in (0,\widehat{y}\hspace{0.1667em}]\end{array}\]
with $M\geqslant 1$, $c_{6}>0$, and $\widehat{y}>0$ defined previously. The lemma is now proved. □3 Proof of Theorem 2
In this section, we prove Theorem 2.
Proof.
Since
\[ \psi (x)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }\Bigg\{{\sum \limits_{i=1}^{n}}(Z_{i}-c\theta _{i})\Bigg\}>x\Bigg),\]
the desired bound of Theorem 2 can be derived from auxiliary Lemma 1.Namely, supposing that r.v.s $Z_{i}-c\theta _{i}$, $i\in \{1,2,\dots \}$, satisfy all conditions of Lemma 1, we get
for all $x\geqslant 0$ with some positive $c_{7}$, $c_{8}$ independent of x.
Therefore,
\[ \psi (x)\leqslant c_{7}{\mathrm{e}}^{-c_{8}x/2}{\mathrm{e}}^{-c_{8}x/2}\leqslant {\mathrm{e}}^{-c_{8}x/2},\]
with $x\geqslant \max \{0,(2\ln c_{7})/c_{8}\}$,Thus, it suffices to check all three assumptions in our lemma with random variables $Z_{i}-c\theta _{i},\hspace{0.1667em}i\in \mathbb{N}$. The requirement ($\mathcal{C}{3}^{\ast }$) of Lemma 1 is evidently satisfied by condition ($\mathcal{C}$3).
Next, it follows from ($\mathcal{C}$1) that
So, the requirement ($\mathcal{C}{1}^{\ast }$) holds too.
It remains to prove that
To establish this, we use the inequality
(11)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|Z_{i}-c\theta _{i}|\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\big)& \displaystyle \leqslant \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\big)\\{} & \displaystyle \hspace{1em}+c\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(\theta _{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\big).\end{array}\]Taking the limit as $u\to \infty $ in the first summand of the right side of inequality (11), we get
(12)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}\leqslant \frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{2em}+\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}\leqslant -u/2\}}\big)\\{} & \displaystyle \hspace{2em}+\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}=\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}=\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}Z_{i}\mathbb{P}\Big(\theta _{i}>\frac{u}{2c}\Big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{i\in \mathbb{N}}{\sup }\mathbb{E}Z_{i}\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{P}\Big(\theta _{i}>\frac{u}{2c}\Big).\end{array}\]Since $x\leqslant {\mathrm{e}}^{\gamma x}/\gamma $, $x\geqslant 0$, condition ($\mathcal{C}$1) implies that
In addition,
by condition ($\mathcal{C}$2).
(14)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{P}\Big(\theta _{i}>\frac{u}{2c}\Big)& \displaystyle =\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\bigg(\frac{\theta _{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}}{\theta _{i}}\bigg)\\{} & \displaystyle \leqslant \underset{u\to \infty }{\lim }\frac{2c}{u}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(\theta _{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)=0\end{array}\]Now taking the limit as $u\to \infty $ in the second summand of the right side of inequality (11), by condition ($\mathcal{C}$2) we have
We now see that the desired equality (10) follows from (11), (15), and (16). This means that all requirements of Lemma 1 hold for r.v.s $Z_{i}-c\theta _{i}$, $i\in \mathbb{N}$. □