Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

Let $ mathcal{T} $ be a positive random variable independent of a real-valued stochastic process $ left{ {X(t),tgeqslant 0} right} $ . In this paper, we investigate the asymptotic behavior of $ mathrm{P}left( {{sup_{{tin left[ {0,mathcal{T}} right]}}}X(t)>u} right) $ as u→∞ assuming that X is a strongly dependent stationary Gaussian process and $ mathcal{T} $ has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that $ mathrm{P}left( {{sup_{{tin left[ {0,mathcal{T}} right]}}}X(t)>u} right)={c^{lambda }}mathrm{P}left( {mathcal{T}>m(u)} right)left( {1+o(1)} right) $ with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.