Extremes of Gaussian processes with a smooth random trend

Vladimir Piterbarg, Goran Popivoda, Siniša Stamatović

Let ξ(t), t ∈ R, be a Gaussian zero mean stationary process, and η(t) another random process, smooth enough, being independent of ξ(t). We will consider the process ξ(t) + η(t) such that conditioned on η(t) it is a Gaussian process. We want to establish an asymptotic exact result for P( sup(E(t))>u), as u → ∞, where T > 0.