Some results for extreme value processes in analogy to the Gaussian spectral representation

The extremal coefficient function has been discussed as an analog of the autocovariance function for extreme values. However, as to the behavior of valid extremal coefficient functions little is known apart from their positive definite type. In particular, the reconstruction of valid processes from given extremal coefficient functions has not been considered before. We show, for the one-dimensional case, the equivalence of the set correlation functions and the extremal coefficient functions with finite range on a grid, and study an analogy to Bochner's theorem, namely that any such extremal coefficient function is representable as a convex combination of a finite set of positive definite functions. This allows for the construction of simple max-stable processes complying with a given extremal coefficient function and, in addition, highlights further properties of the latter. We will include an application of this approach and discuss several examples. As to processes with infinite range we will consider a natural extension of the term "long memory" that is well-known in the Gaussian framework to max-stable processes.