• Bernoulli
  • Volume 21, Number 3 (2015), 1575-1599.

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

Takashi Owada and Gennady Samorodnitsky

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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.

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Bernoulli, Volume 21, Number 3 (2015), 1575-1599.

Received: July 2013
Revised: December 2013
First available in Project Euclid: 27 May 2015

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conservative flow extreme value theory pointwise dual ergodicity sample maxima stable process


Owada, Takashi; Samorodnitsky, Gennady. Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21 (2015), no. 3, 1575--1599. doi:10.3150/14-BEJ614.

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