On Exceedance Point Processes for Stationary Sequences under Mild Oscillation Restrictions

Abstract

It is known ([1]) that any point process limit for the (time normalized) exceedances of high levels by a stationary sequence is necessarily Compound Poisson, under general dependence restrictions. This results from the clustering of exceedances where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes.

Here we investigate a class of stationary sequences satisfying a mild local dependence condition restricting the extent of local “rapid oscillation”. For this class, criteria are given for the existence and value of the so-called “extremal index” which plays a key role in determining the intensity of cluster positions. Cluster size distributions are investigated for this class and in particular shown to be asymptotically equivalent to those for lengths of runs of consecutive exceedances above the level. Relations between the point processes of exceedances, cluster centers, and upcrossings are discussed.