Limit theorems for local cumulative shock models with cluster shock structure
2015
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.
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