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

We study the exact asymptotics of P(supt∈[0,S]X(t)>u)P(supt∈[0,S]X(t)>u), as u→∞u→∞, for centered Gaussian processes with the covariance function satisfying 1−Cov(X(t),X(t+h))=A(t)|h|α(t)+o(|h|α(t)), as h→0h→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 α(t)≡αα(t)≡α. It appears that the behavior of α(t)α(t) in the neighborhood of its global minimum on [0,S][0,S] significantly influences the asymptotics.As an illustration we work out the case of X(t)X(t) being a standardized multifractional Brownian motion.