The supremum of a Gaussian process over a random interval

Abstract

The aim of this note is to give the exact asymptotics of P sup s∈[0,T] X(s)>u as u→∞, where {X(t) : t⩾0} is a centered Gaussian process with stationary increments and T is an independent non-negative random variable with regularly varying tail distribution. In addition, we obtain explicit lower and upper bounds for the prefactor. As an example we analyze the case of X( t) being a fractional Brownian motion and a Gaussian integrated process.