STRONG CONVERGENCE OF MULTIVARIATE POINT PROCESSES OF EXCEEDANCES

E. KAUFMANN AND R.-D. REISS

FB 6, Universität Gesamthochschule Siegen, Hölderlinstr. 3, D-57068 Siegen, Germany

(Received December 24, 1991; revised October 26, 1992)

Abstract. We study the asymptotic behavior of vectors of point processes of exceedances of random thresholds based on a triangular scheme of random vectors. Multivariate maxima w.r.t. marginal ordering may be regarded as a special case. It is proven that strong convergence—that is convergence of distributions w.r.t. the variational distance—of such multivariate point processes holds if, and only if, strong convergence of multivariate maxima is valid. The limiting process of multivariate point processes of exceedances is built by a certain Poisson process. Auxiliary results concerning upper bounds on the variational distance between vectors of point processes are of interest in its own right.

Key words and phrases: Poisson processes, exceedances, random threshold.