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 Fréchet process. It is a self-similar process with $\alpha$-Fréchet 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.

Published at http://dx.doi.org/10.3150/14-BEJ614 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Stationary max-stable fields associated to negative definite functions
• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes

Cited by in corpus (1)

• Time-changed extremal process as a random sup measure