Time-changed extremal process as a random sup measure
Lacaux, Céline; Samorodnitsky, Gennady
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.
ARO grant W911NF-12-10385 and NSA grant H98230-11-1-0154
extremal process; random sup measure; heavy tails; stable process; extremal limit theorem; stationary max-increments; self-similar process