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

Time-changed extremal process as a random sup measure

Céline Lacaux and Gennady Samorodnitsky

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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

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

Received: October 2014
Revised: February 2015
First available in Project Euclid: 3 May 2016

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extremal limit theorem extremal process heavy tails random sup measure stable process stationary max-increments self-similar process


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.

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