On the excursion random measure of stationary processes

Abstract

The excursion random measure $\zeta$ of a stationary process is defined on sets $E \subset (-\infty, \infty) \times (0, \infty)$, as the time which the process (suitably normalized) spends in the set E . Particular cases thus include a multitude of features (including sojourn times) related to high levels. It is therefore not surprising that a single limit theorem for $\zeta$ at high levels contains a wide variety of useful extremal and high level exceedance results for the stationary process itself.

The theory given for the excursion random measure demonstrates, under very general conditions, its asymptotic infinite divisibility with certain stability and independence of increments properties leading to its asymptotic distribution (Theorem 4.1). The results are illustrated by a number of examples including stable and Gaussian processes.