Extremes of independent stochastic processes: a point process approach

For each n a parts per thousand yen 1, let be independent copies of a nonnegative continuous stochastic process X (n) = (X (n) (s)) (saS) indexed by a compact metric space S. We are interested in the process of partial maxima where the brackets [ a <... ] denote the integer part. Under a regular variation condition on the sequence of processes X (n) , we prove that the partial maxima process weakly converges to a superextremal process as . 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.