Extremes of space–time Gaussian processes

Let Z = { Z t ( h ) ; h ∈ R d , t ∈ R } be a space–time Gaussian process which is stationary in the time variable t . We study M n ( h ) = sup t ∈ [ 0 , n ] Z t ( s n h ) , the supremum of Z taken over t ∈ [ 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 → ∞ , the process b n ( M n − b n ) converges as n → ∞ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.