Limit Theorems for Local Cumulative Shock Models with Cluster Shock Structure

الملخص EN

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 andsecondary shocks.

Each primary shock causes a series of secondary shocks according tosome cluster pattern.

In reliability applications, a primary shock can represent an issueof insurance policies of an insurer company, and the secondary shocks then denote therelevant insurance claims generated by the policy.

We focus on the local cumulative shockprocess where only a certain number of the most recent primary and secondary shocksare accumulated.

This process is a very new topic in the available literature which ismore flexible and realistic in modeling some more complex reliability situations such asbankrupt behavior of an insurance company.

Based on the theory of infinite divisibilityand stable distributions, we establish a central limit theorem for the local cumulativeshock process and obtain the conditions for the process to converge to an infinitely divisibledistribution or to an α-stable law.

Also, by choosing the proper scale parameters, theprocess converges to a normal distribution.