Extremal stochastic integrals: a parallel between max–stable processes and α−stable processes

We construct extremal stochastic integrals ∫ e E f(u)Mα(du) of a deterministic function f(u) ≥ 0 with respect to a random α−Fréchet (α > 0) sup–measure. The measure Mα is sup–additive rather than additive and is defined over a general measure space (E, E , μ), where μ is a deterministic control measure. The extremal integral is constructed in a way similar to the usual α−stable integral, but with the maxima replacing the operation of summation. It is well–defined for arbitrary f(u) ≥ 0, ∫ E f(u)μ(du) <∞, and the metric ρα(f, g) := ∫ E |f(u)−g(u)|μ(du) metrizes the convergence in probability of the resulting integrals. This approach complements the well–known de Haan’s spectral representation of max– stable processes with α−Fréchet marginals. De Haan’s representation can be viewed as the max–stable analog of the LePage series representation of α−stable processes, whereas the extremal integrals correspond to the usual α−stable stochastic integrals. We prove that essentially any strictly α−stable process belongs to the domain of max–stable attraction of an α−Fréchet, max–stable process. Moreover, we express the corresponding α−Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the α−stable process. The close correspondence between the max–stable and α−stable frameworks yields new examples of max–stable processes with non–trivial dependence structures. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS–0505747 at Boston University. AMS Subject classification. Primary 60G70, 60G52 secondary 60E07.