Extremal properties of shot noise processes

Abstract

Consider the shot noise process X(t):= Σih(t – τi), , where h is a bounded positive non-increasing function supported on a finite interval, and the are the points of a renewal process η on [0, ). In this paper, the extremal properties of {X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of {X(t)} at the points of η. The important special case where η is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product.