Extremes of independent stochastic processes: a point process approach

TL;DR

This paper studies the asymptotic behavior of maxima of independent stochastic processes using a point process approach, characterizing the limit process and its properties under regular variation conditions.

Contribution

It introduces a novel point process framework for analyzing the extremes of independent stochastic processes and characterizes the limit superextremal process.

Findings

The partial maxima process converges weakly to a superextremal process.

The limit process is max-stable, self-similar, and satisfies a homogeneous Markov property.

Applications include the extremes of log-normal and Gaussian processes.

Abstract

For each $n\geq 1$, let $ {X_{in}, \quad i \geq 1} $ be independent copies of a nonnegative continuous stochastic process $X_{n}=(X_n(t))_{t\in T}$ 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 $n\to\infty$. 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.