Extremes of space–time Gaussian processes

Let Z={_Zt(h);h∈R^d,t∈R}$Z=\{Z_t\left(h\right);h\in {\mathbb{R}}^d,t\in \mathbb{R}\}$ be a space–time Gaussian process which is stationary in the time variable t$t$. We study _Mn(h)=_{supt∈0,n]Zt}(_snh)$M_n\left(h\right)={sup}_{t\in 0,n]}{Z}_t\left(s_{n}h\right)$, the supremum of Z$Z$ taken over t∈0,n]$t\in 0,n]$ and rescaled by a properly chosen sequence _sn→0$s_n\to 0$. Under appropriate conditions on Z$Z$, we show that for some normalizing sequence _bn→∞$b_n\to \infty$, the process _bn(_Mn−_bn)$b_n(M_n-b_n)$ converges as n→∞$n\to \infty$ to a stationary max-stable process of Brown–Resnick type. Using strong approximation, we derive an analogous result for the empirical process.