Extremes of independent stochastic processes: a point process approach

For each $n\ge 1$, let $X_{in},i\ge 1$ be independent copies of a nonnegative continuous stochastic process $X_n=(X_n(t){)}_{t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima [\tilde M_n(u,t) =\max {X_{in}(t), 1 \leq i\leq [nu]},\quad u\geq 0,
t\in T.] where the brackets $[\cdot ]$ 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 $n\to \mathrm{\infty }$. 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.