Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let ({ X_{in}, quad i geqslant 1 }) be independent copies of a nonnegative continuous stochastic process X _n = (X _n (s)) _s∈S indexed by a compact metric space S. We are interested in the process of partial maxima (tilde M_{n}(t,s) =max { X_{in}(s), 1 leqslant ileqslant [nt] },quad tgeq 0, sin S,) where the brackets [ ? ] denote the integer part. Under a regular variation condition on the sequence of processes X _n , we prove that the partial maxima process (tilde M_{n}) weakly converges to a superextremal process (tilde M) as (nto infty ). 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.