Ergodicity and weak mixing for group-indexed infinitely divisible stationary processes

TL;DR

This paper proves that ergodic infinitely divisible stationary processes indexed by any group are inherently weakly mixing, extending known Gaussian results to a broader class without structural group assumptions.

Contribution

It establishes that ergodicity implies weak mixing for all such processes, removing previous restrictions on the group structure and generalizing the theory.

Findings

Ergodic infinitely divisible stationary processes are weakly mixing.

The result holds for any indexing group, including non-abelian groups.

A new construction of stochastically continuous extensions is introduced.

Abstract

We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing. This shows that, as in the well-known case of Gaussian stationary processes, ergodicity implies weak mixing is intrinsic to infinite divisibility, removing all structural assumptions on the group from prior results. The main ingredient is a general construction of stochastically continuous extensions for separable in probability stationary processes, reducing the problem to stochastically continuous processes indexed by Polish groups and then to countable groups, where we combine the Maruyama representation with an ergodicity criterion for Poisson suspensions.