Interestingly there are two different ways of charactaresing max-stable processes: these are known as the spectral characterisations. An especially useful special case of the characterisation of de Haan [1984], is

Another very useful spectral characterisation for unit Frechet max-stable processes is [Schlather, 2002]

Currently there are several useful models based on Schlather's characterisation. The first model, the Schlather's model, is to use , where is a standard Gaussian process. This leads to the bivariate distribution function