Time-changed extremal process as a random sup measure

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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.

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ARO grant W911NF-12-10385 and NSA grant H98230-11-1-0154
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2014-10-09
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extremal process; random sup measure; heavy tails; stable process; extremal limit theorem; stationary max-increments; self-similar process
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