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.
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The author was supported by the Deutsche Forschungsgemeinschaft.
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Kaufmann, E., Reiss, R.D. Strong convergence of multivariate point processes of exceedances. Ann Inst Stat Math 45, 433–444 (1993). https://doi.org/10.1007/BF00773345
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DOI: https://doi.org/10.1007/BF00773345