ON THE EXCURSION RANDOM MEASURE OF STATIONARY-PROCESSES

The excursion random measure zeta of a stationary process is defined o n sets E subset of (-infinity, infinity) x (0, infinity), as the time which the process (suitably normalized) spends in the set E. Particula r 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 usefu l extremal and high level exceedance results for the stationary proces s itself. The theory given for the excursion random measure demonstrat es, under very general conditions, its asymptotic infinite divisibilit y with certain stability and independence of increments properties lea ding to its asymptotic distribution (Theorem 4.1). The results are ill ustrated by a number of examples including stable and Gaussian process es.