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.

Published at http://dx.doi.org/10.3150/15-BEJ717 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
• Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments

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