Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

Abstract

We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.

Published in at http://dx.doi.org/10.1214/13-AOP899 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Recent advances in invariance principles for stationary sequences
• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
• Ergodic properties of Poissonian ID processes
• Random rewards, fractional Brownian local times and stable self-similar processes

Cited by in corpus (3)

• Time-changed extremal process as a random sup measure
• Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments
• Large deviations for occupation and waiting times of infinite ergodic transformations