# Extremes of independent stochastic processes: a point process approach

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1 LMA-Poitiers - Laboratoire de Mathématiques et Applications

Abstract : For each $n\geq 1$, let $\{ X {in}, \quad i \geq 1 \}$ be independent copies of a nonnegative continuous stochastic process $X {n}=X nt {t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima \ \tilde M nu,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 $n\to\infty$. 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.

Keywords : extreme value theory partial maxima process superextremal process functional regular variations weak convergence

Autor: Clément Dombry - Frédéric Eyi-Minko -

Fuente: https://hal.archives-ouvertes.fr/

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