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On the supremum from gaussian processes over infinite horizon

Dębicki Krzysztof, Michna Zbigniew, Rolski Tomasz

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

83--100

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].