Uniform tail approximation of homogenous functionals of Gaussian fields

TL;DR

This paper develops uniform tail probability approximations for a broad class of functionals of Gaussian fields, extending classical results and providing new bounds and constants relevant for extreme value theory.

Contribution

It introduces uniform approximations for tail probabilities of Gaussian functionals depending on parameters, extending existing asymptotic results to more general settings.

Findings

Derived uniform upper bounds for probabilities of double maxima

Extended Piterbarg-Prisyazhnyuk theorem to new classes of Gaussian functionals

Proved finiteness of generalized Piterbarg constants

Abstract

Let $X(t),t\in R^d$ be a centered Gaussian random field with continuous trajectories and set $\xi_u(t)= X(f(u)t),t\in R^d$ with $f$ some positive function. Classical results establish the tail asymptotics of $P\{ \Gamma(\xi_u) > u\}$ as $u\to \infty$ with $\Gamma(\xi_u)= \sup_{t \in [0,T ]^d} \xi_u(t),T>0$ by requiring that $f(u) \to 0$ with speed controlled by the local behaviour of the correlation function of $X$. Recent research shows that for applications more general continuous functionals than supremum should be considered and the Gaussian field can depend also on some additional parameter $\tau_u \in K$, say $\xi_{u,\tau_u}(t),t\in R^d$. In this contribution we derive uniform approximations of $P\{ \Gamma(\xi_{u,\tau_u})> u\}$ with respect to $\tau_u$ in some index set $K_u$, as $u\to\infty$. Our main result have important theoretical implications; two applications are already included in [10,11]. In this paper we present three additional ones, namely i) we derive uniform upper bounds for the probability of double-maxima, ii) we extend Piterbarg-Prisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields $\xi_{u}$, and iii) we show the finiteness of generalized Piterbarg constants.