Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields

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 $\mathbb{E}[X(t)]=0$. 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 $\rho$.

References in corpus (2)

• Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
• The distribution of maxima of approximately Gaussian random fields

Cited by in corpus (1)

• Uniform tail approximation of homogenous functionals of Gaussian fields