## Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 1566-1598.

### 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 $\mathbb{E}[X(t)]=\mathbf{0}$. For any compact sets $A_{1},A_{2}\subset\mathbb{R}^{N}$, precise asymptotic behavior of the excursion probability $\mathbb{P}(\max_{s\in A_{1}}X_{1}(s)>u,\max_{t\in A_{2}}X_{2}(t)>u)\qquad\mbox{as }u\rightarrow\infty$ 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 $\rho$.

#### Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 1566-1598.

Dates
Revised: October 2015
First available in Project Euclid: 17 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1489737618

Digital Object Identifier
doi:10.3150/15-BEJ788

Mathematical Reviews number (MathSciNet)
MR3624871

Zentralblatt MATH identifier
06714312

#### Citation

Zhou, Yuzhen; Xiao, Yimin. Tail asymptotics for the extremes of bivariate Gaussian random fields. Bernoulli 23 (2017), no. 3, 1566--1598. doi:10.3150/15-BEJ788. https://projecteuclid.org/euclid.bj/1489737618

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