 Research
 Open Access
Web renewal counting processes and their applications in insurance
 Rong Li^{1},
 Xiuchun Bi^{1}Email author and
 Shuguang Zhang^{1}
https://doi.org/10.1186/s1366001818540
© The Author(s) 2018
 Received: 12 January 2018
 Accepted: 13 September 2018
 Published: 25 September 2018
Abstract
This paper investigates a nonstandard renewal counting process with dependent interarrival timesweb renewal process. Several limit properties, including the tail of the exponential moment which is a crucial condition in many situations, are obtained. Then the results are applied in insurance to derive precise large deviations and moderate deviation formulas for the aggregate amount of claims.
Keywords
 Renewal processes
 Dependence
 Web Markov skeleton processes
 Precise large deviations
 Moderate deviations
MSC
 62P05
 62E05
 60K30
 60K05
1 Introduction
Renewal processes are important counting processes and are used in various fields. In this paper, we investigate a nonstandard renewal counting process with nonindependent interarrival times \(T_{1},T_{2},\dots\). The motivation of this paper comes from web Markov skeleton processes (WMSPs for short).
Suppose that \(\lambda(t)=\mathbf {E}N_{t}\rightarrow\infty\) as \(t\rightarrow \infty\). If the interarrival times \(T_{1},T_{2},\dots\) form a sequence of independent identically distributed (i.i.d.) random variables, then (1.4) is the standard renewal process which is an important counting process in many applications, such as renewal risk model in risk theory. Some important limit properties of the standard renewal process, such as convergence and the limit distribution, have been extensively investigated in much literature (see Ross [24], Kaas and Tang [10], Ng et al. [23], Tang and Tsitsiashvili [28], among many others). But there are few results for the corresponding web renewal process. These are what we are going to study.
The rest of the paper is organized as follows. Section 2 gives the main results for \(N_{t}\) and the proofs after some preliminaries. Section 3 presents some applications in insurance and derives the results of precise large deviations and moderate deviations for the web renewal risk process \(S_{t}\). Section 4 concludes this paper. Some proofs are provided in the appendix.
2 Main results and discussions
In this section we will give several limit properties of the counting process \(N_{t}\).
A sequence \(T_{n}, n \geq1\) of random variables is Mdependent, where M is a positive integer, if \(T_{1},T_{2}, \dots, T_{i}\) is independent of \(T_{j},T_{j+1}, \dots\) for \(ji>M\). Now we are in a position to state our main results.
Theorem 2.1
In fact, it is not difficult to derive (2.1) and (2.2) from Theorems 4 and 7 in Korchevsky and Petrov [13], and Ross [24]. We omit it here.
Remark 2.1
The constraint on \(T_{n}, n \geq1\), Mdependent, is for tractability, and is also natural. Consider the dependent structure (1.2), if \(Y_{n}, n \geq1\), is an i.i.d. or weakly dependent sequence, then it is easy to ensure that \(T_{n}, n \geq1\), is a sequence of kdependence under some conditions.
The following condition is crucial for a counting process in most applications.
Assumption 2.1
Kočetova et al. [12] proved that Assumption 2.1 is satisfied for the standard renewal counting process.
Proposition 2.1
(Kočetova et al. [12])
They also considered the applications of their result in insurance mathematics.
In the case of the standard renewal counting process with the finite mean \(\mathbf {E}T=1/\lambda<\infty\), Assumption 2.1 is equivalent to the following assumption.
Assumption 2.2
(Leipus and Šiaulys [14])
Assumption 2.2 is one of the crucial requirements for the counting process \(N_{t}\) in the paper of Leipus and Šiaulys [14]. Furthermore, in this case, Assumption 2.2 implies the following assumption mentioned by Klüppelberg and Mikosch [11].
Assumption 2.3
(Klüppelberg and Mikosch [11])
Assumption 2.3 is an essential condition in their paper.
The importance of the abovementioned statements can also be found in Kaas and Tang [10], Ng et al. [23], Tang and Tsitsiashvili [28], Wang and Wang [29], Shen et al. [25] and others therein.
Fu and Shen [6] proved the following key lemma when they considered moderate deviations for sums of claims in a sizedependent renewal risk model.
Lemma 2.1
(Fu and Shen [6])
It can be seen that the property of Assumption 2.1 is very important. We will prove that Eqs. (2.3) and (2.4) also hold for the web renewal counting process.
Theorem 2.2
 (i)if \(\mathbf {E}T=1/\lambda<\infty\), then
 (i1)holds for every \(a > (k+1)\lambda\) and some \(b > 1\);$$ \lim_{t\rightarrow\infty} \sum_{n>at}b^{n} \Pr (N_{t}\geq n )=0 $$(2.5)
 (i2)holds for any \(\delta>0\), some \(c>0\) and a positive function \(b(t)\) satisfying \(b(t)/t\rightarrow0\) as \(t\rightarrow\infty\).$$ \lim_{t\rightarrow\infty} \sum_{n>\lambda t+\delta b(t)}c^{n} \Pr (N_{t}\geq n )=0 $$(2.6)
 (i1)
 (ii)
Theorem 2.2 is an extension to the Lemma 3.3 in Bi and Zhang [2], which is a key lemma in the proof of their main results.
Proof
We next give the proof of Theorem 2.2.
We obtain (2.5). Equation (2.6) can be proved similarly.
3 Applications in insurance
In this section, we consider some applications of the main results in insurance. We investigate the precise large deviations and moderate deviations formulas for the web renewal risk process (1.3), where the claims \(\{Y_{n}, n\geq1\}\) are identically distributed and nonnegative random variables (r.v.s) with the common distribution function (d.f.) \(F(x)=\Pr(Y\leq x)\) and the finite mean \(\mathbf {E}Y=\mu\), and the interarrival times \(T_{n}, n\geq1\) depended on \(\{Y_{n}, n\geq1\}\) through dependent structure (1.2)

For two positive functions \(f(x)\) and \(g(x)\), we write$$\begin{aligned} &f(x)\sim g(x)\quad \mbox{if } \lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1, \\ &f(x)\lesssim g(x)\quad \mbox{if } \limsup_{x\rightarrow\infty}\frac{f(x)}{g(x)} \leq1. \end{aligned}$$

For two positive bivariate functions \(f(\cdot,\cdot)\) and \(g(\cdot,\cdot)\), we say that \(f(x,t)\lesssim g(x,t)\), as \(t\rightarrow\infty\), holds uniformly in \(x\in\Delta _{t}\neq\emptyset\), if$$\limsup_{t\rightarrow\infty}\sup_{x\in\Delta_{t}}\frac {f(x,t)}{g(x,t)} \leq1. $$

For a distribution function \(F(x)\) with finite mean \(\mu>0\), set \(\overline{F}(x)\equiv1F(x)\) as the corresponding survival function of it.
3.1 Precise large deviations
The precise large deviation of random sums has been extensively investigated in much literature since it was initiated by Klüppelberg and Mikosch [11], for example, Kaas and Tang [10], Tang [27], Lin [17], Baltrūnas et al. [1], Chen and Zhang [5], Liu [18], Li et al. [15], Chen and Yuen [4], Bi and Zhang [2], Shen et al. [25], Yang and Sha [30], Guo et al. [8], Hua et al. [9] and Liu et al. [20], among many others.
Assumption 3.1
Then we have the following result for the web renewal risk process.
Theorem 3.1
Consider the web renewal risk process (1.3) with i.i.d. claims. In addition to Assumption 3.1, suppose that \(F\in \mathcal{C}, \mathbf {E}[T] =1/\lambda\in(0,\infty)\) and \(\operatorname {\mathbf {Var}}[T]<\infty\). Then, for any given \(\delta> 0\), (3.3) holds uniformly for all \(x\geq\delta t\).
If we select \(k=1\) in Assumption 3.1, then we get the result of Theorem 2.1 in Bi and Zhang [2]. We restate it as a corollary of Theorem 3.1.
Corollary 3.1
(Bi and Zhang [2])
Consider the aggregate amount of claims (1.3) with i.i.d. claims sequence. In addition to Assumption 3.2, suppose that \(F\in \mathcal{C}, \mathbf {E}[T] =1/\lambda\in(0,\infty)\) and \(\operatorname {\mathbf {Var}}[T]<\infty\). Then, for any given \(\delta> 0\), (3.3) holds uniformly for all \(x\geq\delta t\).
The proof of Theorem 3.1 is similar to that of Corollary 3.1. We omit it here. Pay attention that Theorem 2.2 is one of the key conditions for the proof and Theorem 3.1 really extends Corollary 3.1.
Remark 3.1
For the mutually independent claims \(\{Y_{n}, n\geq1\}\), it is easy to construct an example that \(T_{n},n\geq1\) is kdependent under the Assumption 3.1. But if \(\{Y_{n}, n\geq1\}\) are not mutually independent, it is hard to construct such sequence of kdependence only under the Assumption 3.1.
If the risks form an extended negative dependent (END) sequence, we need additional conditions to obtain the results of precise large deviations, which will be considered in the next subsection.
3.2 Moderate deviations
Moderate deviations extend precise large deviations through extend xregion. Note that the xregion in Theorem 3.1 is taken as \([\delta t,\infty)\). It is natural to ask whether (3.3) can still hold for \(x\in[\gamma b(t), \infty)\) with \(b(t)/t\rightarrow0\) as \(t\rightarrow\infty\), and if it can, what conditions are appropriate. Similar problems were partly studied by Shen and Zhang [26] for a risk model based on the customerarrival process, by Gao [7] and Liu [19] for the standard renewal risk model with independent and dependent claims, respectively and by Fu and Shen [6] for the sums of consistently varying tailed claims in a sizedependent renewal risk model.
Taking Theorem 2.2 into consideration, we obtain the following result for the web renewal risk process proposed above.
Theorem 3.2
Proof
See the Appendix. □
Remark 3.2

\(a(t)< C t\) for t large enough and a positive constant C;

\(\lim_{t\rightarrow\infty}\frac{a(t)}{a([t])}=1\);

\(\lim_{t\rightarrow\infty}\frac{n(\log n)^{\alpha }}{a(n)^{\alpha}}=0, 1<\alpha< \min\{2, J^{+}_{F}\}\),
Taking \(a(n)=n^{1/\alpha}(\log n)^{2}\), then \(a(n)\) satisfies the conditions in Remark 3.2 and \(b(t)/t\rightarrow0\) as \(t\rightarrow\infty\).
Theorem 3.2 is an extension to Theorem 2.1 in Fu and Shen [6]. Furthermore, we have
Theorem 3.3
Consider the renewal risk process (1.3) with END risk sequence. In addition to Assumption 3.1 with \(\mathbf {E}T^{*} <\infty\), suppose that \(F\in\mathcal{C}\), \(\mathbf {E}T=1/\lambda<\infty\), \(\mathbf {E}Y^{\beta}<\infty\) for some \(\beta>\alpha>1\) and \(\operatorname {\mathbf {Var}}[T]<\infty \). If (3.7) holds, then, for any given \(\gamma>0\), (3.8) holds uniformly for all \(x\geq\gamma b(t)\), where \(b(t)=a(\lambda t)\) is a positive function satisfying the conditions in Theorem 3.2.
Then we get the following result for precise large deviations.
Corollary 3.2
Consider the renewal risk process (1.3) with END risk sequence. In addition to Assumption 3.1 with \(\mathbf {E}T^{*} <\infty\), suppose that \(F\in\mathcal{C}\), \(\mathbf {E}T=1/\lambda<\infty\), \(\mathbf {E}Y^{\beta}<\infty\) for some \(\beta>1\) and \(\operatorname {\mathbf {Var}}[T]<\infty\). then, for any given \(\gamma>0\), (3.8) holds uniformly for all \(x\geq\gamma t\).
The proof of Theorem 3.3 is similar to that of Theorem 3.2.
Remark 3.3
For the END claims \(\{Y_{n}, n\geq1\}\), in addition to Assumption 3.1, one needs additional conditions to ensure the sequence \(\{T_{n}, n\geq1\}\) to be kdependent.
4 Conclusions
Motivated by Ma et al. [22], this paper investigates a nonstandard renewal counting process with kdependent interarrival times, and obtains some important limit properties. We obtained the tail of the exponential moment of the counting process, which is crucial in many situations. We considered the applications of the main results in risk theory, and derived the formulas of precise large deviations and moderate deviations of the web renewal risk process. These results allow applications in various natural and social sciences.
Many topics based on the web renewal risk process shall be investigated. For example, Li et al. [16] studied a stochastic interest model based on compound Poisson process, and it is of interest to study the problem based on the web renewal risk process.
5 Methods/experimental
Not applicable.
Declarations
Acknowledgements
The authors are grateful to the editor and the anonymous referees for conscientious comments and corrections.
Availability of data and materials
Not applicable.
Funding
This work was Supported by National Natural Science Foundation of China (grant numbers 11401556, 11471304).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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