On the supremum from gaussian processes over infinite horizon

Abstract: In the paper we study the asymptotic of the tail of distribution function P (A(X, c) > x) for x → ∞, where A(X, c) is the supremum of X(t)−ct over [0,∞). In particular, X(t) is the fractional Brownian motion, a nonlinearly scaled Brownian motion or some integrated stationary Gaussian processes. For the fractional Brownian motion we give a stronger result than a recent one of Duffield and O’Connell [5].