Extremes of independent stochastic processes: a point process approach

Frédéric Eyi-Minko (1) , Dombry Clément (2)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

For each $n\geq 1$, let $\{X_i, 1\leq i\leq n \}$ 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 $\tilde M_n (t, s) = \max\{X_i(s), 1\leq i\leq [nt]\}, t\geq 0, S\in S$ 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 $n\to\infty$. 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.

Mots clés: Extreme value theory · Partial maxima process · Superextremal; process · Functional regular variations · Weak convergence; AMS 2000 Subject Classifications Primary: 60G70 · Secondary: 60F17; extreme value theory; partial maxima process; superextremal process; functional regular variations; Weak convergence of Markov processes

Domaines: Probabilités [math.PR]

Frédéric Eyi-Minko, Dombry Clément. Extremes of independent stochastic processes: a point process approach. Extremes, 2016, 19, pp.197 - 218. ⟨10.1007/s10687-016-0243-7⟩. ⟨hal-01771205⟩