Time-changed extremal process as a random sup measure

TL;DR

This paper introduces a new class of self-similar Fréchet processes with stationary max-increments, derived from a time-changed extremal process, revealing intricate self-affine structures as limits of long memory stable sequences.

Contribution

It uncovers the self-affine structure of a time-changed extremal process as a random sup measure and links it to limits of long memory stable sequences, expanding the class of known extremal processes.

Findings

The limiting process is described as a $eta$-power time change of the classical Fréchet extremal process.

The resulting process has stationary max-increments and self-affine properties.

A new class of self-similar Fréchet processes with stationary max-increments is constructed.

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 Fr\'echet 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 Fr\'echet processes with stationary max-increments.