Extremes of Gaussian Processes with a Smooth Random Trend

Abstract

Let $\xi(t)$, $t\in\mathbf{R}$, be a Gaussian zero mean stationary process, and $\eta(t)$ another random process, smooth enough, being independent of $\xi(t)$. We will consider the process $\xi(t)+\eta(t)$ such that conditioned on $\eta(t)$ it is a Gaussian process. We want to establish an asymptotic exact result for
$$\vjer\left(\sup_{t\in [0,T]} (\xi(t)+\eta(t))>u\right),\text{ as }u\to\infty,$$
where $T>0$.