Maxima of Long Memory Stationary Symmetric Α-stable Processes, and Self-similar Processes with Stationary Max-increments

We derive a functional limit theorem for the partial maxima process based on a long memory stationary α-stable 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 Fréchet process. It is a self-similar process with α-Fréchet marginals, and it has the property of stationary max-increments, which we introduce in this paper. The functional limit theorem is established in the space D[0,∞) equipped with the Skorohod M1topology; in certain special cases the topology can be strengthened to the Skorohod J1-topology.