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

Abstract

Let \( \mathcal{T} \) be a positive random variable independent of a real-valued stochastic process \( \left\{ {X(t),t\geqslant 0} \right\} \). In this paper, we investigate the asymptotic behavior of \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) \) 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 \( \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) \) 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.