## Bernoulli

• Bernoulli
• Volume 22, Number 4 (2016), 1979-2000.

### Time-changed extremal process as a random sup measure

#### Abstract

A functional limit theorem for the partial maxima of a long memory stable sequence produces a limiting process that can be described as a $\beta$-power time change in the classical Fréchet extremal process, for $\beta$ in a subinterval of the unit interval. Any such power time change in the extremal process for $0<\beta<1$ produces a process with stationary max-increments. This deceptively simple time change hides the much more delicate structure of the resulting process as a self-affine random sup measure. We uncover this structure and show that in a certain range of the parameters this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. These results open a way to construct a whole new class of self-similar Fréchet processes with stationary max-increments.

#### Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 1979-2000.

Dates
Revised: February 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297672

Digital Object Identifier
doi:10.3150/15-BEJ717

Mathematical Reviews number (MathSciNet)
MR3498020

Zentralblatt MATH identifier
1346.60070

#### Citation

Lacaux, Céline; Samorodnitsky, Gennady. Time-changed extremal process as a random sup measure. Bernoulli 22 (2016), no. 4, 1979--2000. doi:10.3150/15-BEJ717. https://projecteuclid.org/euclid.bj/1462297672

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