Asymptotics of supremum distribution of [alpha](t)-locally stationary Gaussian processes

We study the exact asymptotics of , as u-->[infinity], for centered Gaussian processes with the covariance function satisfying 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)[reverse not equivalent][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.