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

TL;DR

This paper proves a new functional central limit theorem for heavy-tailed stationary infinitely divisible processes, linking the limit behavior to ergodic properties of the underlying flow, and introduces a new class of stable self-similar processes.

Contribution

It introduces a novel class of functional CLTs for heavy-tailed processes generated by conservative flows, connecting limit processes to ergodic-theoretical properties.

Findings

Limit processes are symmetric stable self-similar with stationary increments.

Normalizing sequences depend on ergodic properties of the underlying flow.

Functional convergence is established for conservative pointwise dual ergodic maps.

Abstract

We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying L\'{e}vy 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.