Maxima of long memory stationary symmetric alpha-stable processes, and self-similar processes with stationary max-increments

摘要

We derive a functional limit theorem for the partial maxima process based on a long memory stationary astable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Frechet process. It is a self-similar process with alpha-Frechet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space D[0, infinity) equipped with the Skorohod M-1-topology; in certain special cases the topology can be strengthened to the Skorohod J(1)-topology.