Asymptotics of supremum distribution of α(t)-locally stationary Gaussian processes

We study the exact asymptotics of P(sup(t is an element of[0,S]) X (t) > u), as u -> infinity, for centered Gaussian processes with the covariance function satisfying 1 - Cov (X (t), X (t+h)) = A (t)vertical bar h vertical bar alpha((t)) + o (vertical bar h vertical bar alpha((t))), as h -> 0. The obtained results complement those already considered in the literature for the case of locally stationary Gaussian processes in the sense of Berman, where alpha(t) equivalent to alpha. It appears that the behavior of alpha(t) in the neighborhood of its global minimum on [0, S] significantly influences the asymptotics. As an illustration we work out the case of X(t) being a standardized multifractional Brownian motion. (C) 2007 Elsevier B.V. All rights reserved.