On the Multivariate Extremal Index

The exceedance point process approach of Hsing et al. is extended to multivariate stationary sequences and some weak convergence results are obtained. It is well known that under general mixing assumptions, high level exceedances typically have a limiting Compound Poisson structure where multiple events are caused by the clustering of exceedances. In this paper we explore (a) the precise effect of such clustering on the limit, and (b) the relationship between point process convergence and the limiting behavior of maxima. Following this, the notion of multivariate extremal index is introduced which is shown to have properties analogous to its univariate counterpart. Two examples of bivariate moving average sequences are presented for which the extremal index is calculated in some special cases.