The supremum of a Gaussian process over a random interval

The aim of this note is to give the exact asymptotics of P (sup(sis an element of{[}1,T]) X(s) > u) as u --> infinity, where {X(t): t greater than or equal to 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. (C) 2003 Published by Elsevier B.V.