Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields

TL;DR

This paper derives precise asymptotic probabilities for the joint extremes of a bivariate Gaussian random field, considering smoothness and correlation, as the threshold tends to infinity.

Contribution

It provides explicit asymptotic formulas for joint excursion probabilities of bivariate Gaussian fields, incorporating smoothness and correlation effects.

Findings

Asymptotic behavior depends on smoothness parameters and maximum correlation.

Explicit formulas for joint tail probabilities are derived.

Results apply to locally stationary Gaussian fields in high-threshold regimes.

Abstract

Let $\{X(t)= (X_1(t),X_2(t))^T,\ t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued continuous locally stationary Gaussian random field with $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_1, A_2 \subset \mathbb{R}^N$, precise asymptotic behavior of the excursion probability \[ \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty \] is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $\rho$.