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

Abstract

We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-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 Fr\'{e}chet process. It is a self-similar process with $\alpha$-Fr\'{e}chet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_1$-topology.