Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ 0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.