Ergodic properties of Poissonian ID processes

TL;DR

This paper classifies stationary infinitely divisible processes without Gaussian parts into four classes based on their Lévy measures, each exhibiting distinct ergodic properties, and uses Poisson measure representations to analyze these properties.

Contribution

It provides a novel decomposition of stationary IDp processes into four classes with distinct ergodic behaviors, using Poisson measure representations.

Findings

Processes can be decomposed into four classes with specific ergodic properties

Each class exhibits a different ergodic behavior: nonergodicity, weak mixing, mixing, Bernoullicity

Poisson measure representations are key to analyzing ergodic properties

Abstract

We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its L\'{e}vy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects.