Max-stable processes for modeling extremes observed in space and time

Max-stable processes have proved to be useful for the statistical modeling of spatial extremes. For statistical inference it is often assumed that there is no temporal dependence; i.e., that the observations at spatial locations are independent in time. In a first approach we construct max-stable space–time processes as limits of rescaled pointwise maxima of independent Gaussian processes, where the space–time covariance functions satisfy weak regularity conditions. This leads to so-called Brown–Resnick processes. In a second approach, we extend Smith’s storm profile model to a space–time setting. We provide explicit expressions for the bivariate distribution functions, which are equal under appropriate choice of the parameters. We also show how the space–time covariance function of the underlying Gaussian process can be interpreted in terms of the tail dependence function in the limiting max-stable space–time process.