Extremes of space–time Gaussian processes

Let Z={Zt(h);h∈Rd,t∈R}Z={Zt(h);h∈Rd,t∈R} be a space–time Gaussian process which is stationary in the time variable tt. We study Mn(h)=supt∈[0,n]Zt(snh)Mn(h)=supt∈[0,n]Zt(snh), the supremum of ZZ taken over t∈[0,n]t∈[0,n] and rescaled by a properly chosen sequence sn→0sn→0. Under appropriate conditions on ZZ, we show that for some normalizing sequence bn→∞bn→∞, the process bn(Mn−bn)bn(Mn−bn) converges as n→∞n→∞ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.