A note on LDP for supremum of Gaussian processes over infinite horizon

The aim of this paper is to give a short proof of a large deviation result for supremum of nencentered Gaussian process over infinite horizon. We study family {mu(X,d;u); u > 0} of Borel probability measures on R, where mu(X,d;u)(B) = P ((sup)(t greater than or equal to 0) 1/u(X(t) - d(t)) is an element of B) for Borel B subset of R, drift function d(t) and centered Gaussian processes {X (t); t greater than or equal to 0} with variance function sigma(2)(t). We assume that for each 0 less than or equal to epsilon less than or equal to 1 P ((sup)(t greater than or equal to 0) (X(t) - epsilon d(t)) > u ) --> 0 for u --> infinity. We obtain logarithmic asymptotic of P(sup(t greater than or equal to 0) (X(t)) > u). Under additional assumption, that sigma(2)(t) is regularly varying at infinity and d(t) is linear, we prove large deviation principle for {mu(X,d;u); u > 0}. (C) 1999 Elsevier Science B.V. All rights reserved. MSG. primary 60G15; secondary 60G70; 68M20.