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On the multivariate extremal index. (English) Zbl 0881.60050
Summary: The exceedance point process approach of T. Hsing, J. Hüsler and M. R. Leadbetter [Probab. Theory Relat. Fields 78, No. 1, 97-112 (1988; Zbl 0619.60054)] 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. 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.

MSC:
60G70 Extreme value theory; extremal stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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