## Bernoulli

• 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

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.bj/1432732030

Digital Object Identifier
doi:10.3150/14-BEJ614

Mathematical Reviews number (MathSciNet)
MR3352054

Zentralblatt MATH identifier
1325.60040

#### Citation

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. https://projecteuclid.org/euclid.bj/1432732030

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