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STRONG CONVERGENCE OF MULTIVARIATE POINT

PROCESSES OF EXCEEDANCES

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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.

**Source**
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