Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields

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 $$\mathbb{P}(\underset{s\in A_1}{\max }{X}_1(s)>u,\underset{t\in A_2}{\max }{X}_2(t)>u),\text{ as }u\to \mathrm{\infty }$$ 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 $\rho$.