Extremes of space-time Gaussian processes

Let Z = {Z(t) (h); h is an element of R(d), t is an element of R} be a space-time Gaussian process which is stationary in the time variable t. We study M(n)(h) = sup(t is an element of[0, n]) Z(t) (s(n)h), the supremum of Z taken over t is an element of [0, n] and rescaled by a properly chosen sequence s(n) -> 0. Under appropriate conditions on Z, we show that for some normalizing sequence b(n) -> infinity, the process b(n)(M(n) - b(n)) converges as n -> infinity to a stationary max-stable process of Brown-Resnick type. Using strong approximation, we derive an analogous result for the empirical process. (C) 2009 Elsevier B.V. All rights reserved.