Uniform tail approximation of homogenous functionals of Gaussian fields

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.

25 pages

References in corpus (3)

• On Extremal Index of max-stable stationary processes
• Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields
• Extremes of Gaussian Random Fields with regularly varying dependence structure

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