Extremes of independent stochastic processes: a point process approach

Abstract

For each n ≥ 1, let {X in , i1} be independent copies of a nonnegative continuous stochastic process X n = (X n (t)) t∈T indexed by a compact metric space T . We are interested in the process of partial maximãwhere the brackets [ · ] denote the integer part. Under a regular variation condition on the sequence of processes X n , we prove that the partial maxima processM n weakly converges to a superextremal processM as n → ∞. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

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Hidden regular variation for point processes and the single/multiple large point heuristic

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