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.