Abstract
This paper investigates a nonstandard renewal counting process with dependent inter-arrival times-web 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.
Highlights
Renewal processes are important counting processes and are used in various fields
We investigate a nonstandard renewal counting process with nonindependent interarrival times T1, T2, . . . . The motivation of this paper comes from web Markov skeleton processes (WMSPs for short)
We investigate the precise large deviations and moderate deviations formulas for the web renewal risk process (1.3), where the claims {Yn, n ≥ 1} are identically distributed and nonnegative random variables (r.v.s) with the common distribution function (d.f.) F(x) = Pr(Y ≤ x) and the finite mean EY = μ, and the inter-arrival times Tn, n ≥ 1 depended on {Yn, n ≥ 1} through dependent structure (1.2)
Summary
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 T1, T2, . . . . The motivation of this paper comes from web Markov skeleton processes (WMSPs for short). We investigate a nonstandard renewal counting process with nonindependent interarrival times T1, T2, . Let E be a collection of insurance policies, X = {Xn, n ≥ 0} describe the transition behaviors of claims between policies, which forms a Markov chain with state space E, {Yn, n ≥ 1} and T = {Tn, n ≥ 1} represent claims sequence and inter-arrive times of claims, respectively. Form a sequence of independent identically distributed (i.i.d.) random variables, (1.4) is the standard renewal process which is an important counting process in many applications, such as renewal risk model in risk theory.
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