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Mathematical Problems in Engineering
Volume 2015, Article ID 828979, 10 pages
http://dx.doi.org/10.1155/2015/828979
Research Article

Limit Theorems for Local Cumulative Shock Models with Cluster Shock Structure

School of Management, Lanzhou University, Lanzhou 730000, China

Received 29 November 2014; Revised 12 February 2015; Accepted 27 February 2015

Academic Editor: Daniela Boso

Copyright © 2015 Jianming Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers a more general shock model with insurance and financial risk background, in which the system is subject to two types of shocks called primary shocks and secondary shocks. Each primary shock causes a series of secondary shocks according to some cluster pattern. In reliability applications, a primary shock can represent an issue of insurance policies of an insurer company, and the secondary shocks then denote the relevant insurance claims generated by the policy. We focus on the local cumulative shock process where only a certain number of the most recent primary and secondary shocks are accumulated. This process is a very new topic in the available literature which is more flexible and realistic in modeling some more complex reliability situations such as bankrupt behavior of an insurance company. Based on the theory of infinite divisibility and stable distributions, we establish a central limit theorem for the local cumulative shock process and obtain the conditions for the process to converge to an infinitely divisible distribution or to an -stable law. Also, by choosing the proper scale parameters, the process converges to a normal distribution.