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

Nachi Avraham-Re'em, Emmanuel Roy

TL;DR

The paper proves that ergodicity implies weak mixing for every infinitely divisible stationary process indexed by an arbitrary group, provided the process is separable in probability. The authors introduce stochastically continuous extensions to Polish groups, reducing the problem to processes indexed by Polish groups and then by countable subgroups, thereby avoiding structural restrictions on the index group. Central tools include the Maruyama representation of idp processes and the ergodicity characterization of Poisson suspensions, which together show that an ergodic idp process is a factor of a weakly mixing Poisson suspension. This yields a unification with the Gaussian case and removes prior group-related assumptions, significantly broadening the scope of ergodicity implying weak mixing in infinite divisibility contexts.

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.

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

TL;DR

The paper proves that ergodicity implies weak mixing for every infinitely divisible stationary process indexed by an arbitrary group, provided the process is separable in probability. The authors introduce stochastically continuous extensions to Polish groups, reducing the problem to processes indexed by Polish groups and then by countable subgroups, thereby avoiding structural restrictions on the index group. Central tools include the Maruyama representation of idp processes and the ergodicity characterization of Poisson suspensions, which together show that an ergodic idp process is a factor of a weakly mixing Poisson suspension. This yields a unification with the Gaussian case and removes prior group-related assumptions, significantly broadening the scope of ergodicity implying weak mixing in infinite divisibility contexts.

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.
Paper Structure (17 sections, 12 theorems, 93 equations)

This paper contains 17 sections, 12 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Stationary processes, ergodicity, weak mixing
  3. Stationary $G$-processes
  4. Ergodicity and weak mixing
  5. Stochastically continuous extensions
  6. Lost in translation: stationary processes and ergodic theory
  7. Separability in probability
  8. Stochastic continuity
  9. Proof of Theorem \ref{['thm:stochext']}
  10. Proof of the main theorem
  11. Necessary background
  12. Infinitely divisible Poissonian stationary processes
  13. Poisson random measures and stochastic integral
  14. The Maruyama representation
  15. Proof of the main theorem
  16. ...and 2 more sections

Key Result

Lemma 2.3

Let $G$ be a group, and let $\mathbf{X}$ and $\mathbf{Y}$ be independent stationary $G$-processes whose finite dimensional characteristic functions never vanish. If $\mathbf{X}\oplus\mathbf{Y}$ is ergodic then both $\mathbf{X}$ and $\mathbf{Y}$ are ergodic.

Theorems & Definitions (29)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3: Samorodnitsky
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • proof
  • Definition 3.5
  • ...and 19 more