Extremes of Independent Gaussian Processes

TL;DR

This paper characterizes the limiting behavior of normalized maxima and minima of independent Gaussian processes, providing a comprehensive description of their possible limit processes as the number of processes grows large.

Contribution

It offers a complete characterization of the limit processes for maxima and minima of independent Gaussian processes, extending extreme value theory to this setting.

Findings

Identifies all possible limit processes for maxima of Gaussian processes.

Provides analogous results for minima of absolute Gaussian processes.

Establishes conditions for convergence of normalized maxima and minima.

Abstract

For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes which can be obtained as limits, as $n\to\infty$, of the process $a_n(M_n-b_n)$, where $M_n(t)=\max_{i=1,...,n} X_{in}(t)$ and $a_n, b_n$ are normalizing constants. We also provide an analogous characterization for the limits of the process $a_nL_n$, where $L_n(t)=\min_{i=1,...,n} |X_{in}(t)|$.