Asymptotics of maxima of strongly dependent Gaussian processes

TL;DR

This paper investigates the asymptotic behavior of the maximum absolute value of a sequence of strongly dependent stationary Gaussian processes over increasing time intervals, extending previous results to broader dependence conditions.

Contribution

It extends the limit distribution results for maxima of Gaussian processes to include cases with local and long-range strong dependence.

Findings

Established limit distribution for maxima under strong dependence

Extended previous results by Seleznjev (1991)

Applicable to processes with complex dependence structures

Abstract

Let $\{X_{n}(t), t\in[0,\infty)\}, n\in\mathbb{N}$ be a sequence of centered dependent stationary Gaussian processes. The limit distribution of $\sup_{t\in[0,T(n)]}|X_{n}(t)|$ is established as $r_{n}(t)$, the correlation function of $X_{n}$ satisfies the local and long range strong dependence conditions, which extends the results obtained by Seleznjev (1991).