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.
Citation
Takashi Owada. Gennady Samorodnitsky. "Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments." Bernoulli 21 (3) 1575 - 1599, August 2015. https://doi.org/10.3150/14-BEJ614
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