(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.