arXiv:2601.17988v2

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

Nachi Avraham-Re'em, Emmanuel Roy

math.PRmath.DS60G1060E0760G5260G5560H0537A50

Paper metadata

Original submission: January 25, 2026
Latest revision: July 16, 2026 · version 2
Claimed MSC: 60G10, 60E07, 60G52, 60G55, 60H05, 37A50
arXiv categories: math.PR, math.DS
Paper license: CC BY 4.0 ↗
Comments: 23 pages. Added a new section on applications. Accepted in PTRF

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.

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Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The main implication is supported in both stages of generality. The countable-group case follows from the Lévy–Maruyama decomposition: ergodicity rules out positive finite invariant components of the Lévy base, making the Poisson suspension weakly mixing, while the Gaussian component has no nontrivial finite-dimensional spectral part. The separability reduction then transfers this result from a countable dense subgroup to the original topological group.

Main TheoremCorrect

Ergodicity implies weak mixing for separable infinitely divisible stationary processes

Pages 2–3 and 13–15 · Main Theorem and its final proof · arXiv:2601.17988v2

For a separable infinitely divisible stationary process indexed by an arbitrary topological group, ergodicity of the shift action implies weak mixing. The statement covers the Gaussian and non-Gaussian Lévy components simultaneously and does not assume countability of the original index group. Separability permits reduction to a countable dense subgroup without changing the generated probability sigma-field. Infinite divisibility passes to this restriction, and weak mixing for the restricted action implies weak mixing for the original action because every original invariant vector is also invariant under the dense subgroup closure.

Countable-group caseCorrect

Poissonian and Gaussian components

Pages 5–13 · Theorem 3.1 and Section 4 · arXiv:2601.17988v2

The infinitely divisible law splits into deterministic, Gaussian, and Poissonian factors. Ergodicity eliminates the deterministic invariant term. A finite invariant set of positive Lévy measure would make the number of Poisson points in it a nonconstant invariant variable, so no such set exists; the base is null and its Poisson suspension is weakly mixing. For the Gaussian factor, absence of invariant variables rules out atoms in the spectral measure, which is equivalent to weak mixing.

General group caseCorrect

Reduction through a countable dense subgroup

Pages 13–15 · Lemma 4.2 and final proof · arXiv:2601.17988v2

Separability of the process yields a countable set of coordinates generating its sigma-field and a countable subgroup whose action is dense in the original shift representation. The automorphism representation is continuous after passing to the standard realization. Invariant functions for the subgroup and its diagonal action are therefore invariant for the closure, and conversely. Ergodicity, weak mixing, and infinite divisibility are preserved under restriction, so the countable theorem applies and transfers back.

02Proofs3 reported findingsCorrect

Theorems 3.1, Lemma 4.2, and the final proof. A standard-probability realization makes the shift representation continuous in the weak automorphism topology. A countable dense subgroup has the same closed action and hence the same invariant sigma-fields for the action and its diagonal square. For the countable model, the Maruyama representation decomposes the process into a Gaussian part and a Poisson integral over its Lévy measure. Any finite invariant component of that Lévy measure would create a nontrivial invariant event in the compound-Poisson factor. Ergodicity therefore forces the base action to be null, whose Poisson suspension is weakly mixing; the Gaussian component satisfies the classical ergodic-implies-weakly-mixing criterion.

Stochastic extension and Maruyama representationCorrect and complete

Theorems 3.1, Lemma 4.2, and the final proof

Pages 5–15 · Sections 3–4 · arXiv:2601.17988v2

A standard-probability realization makes the shift representation continuous in the weak automorphism topology. A countable dense subgroup has the same closed action and hence the same invariant sigma-fields for the action and its diagonal square. For the countable model, the Maruyama representation decomposes the process into a Gaussian part and a Poisson integral over its Lévy measure. Any finite invariant component of that Lévy measure would create a nontrivial invariant event in the compound-Poisson factor. Ergodicity therefore forces the base action to be null, whose Poisson suspension is weakly mixing; the Gaussian component satisfies the classical ergodic-implies-weakly-mixing criterion.

Theorem 3.1Correct and complete

Stochastic realization and continuity of the action

Pages 5–9 · Theorem 3.1 · arXiv:2601.17988v2

The coordinate process is realized on a standard probability space generated by a countable separating family. Equality of finite-dimensional distributions defines the shift automorphisms, and convergence on the generating family gives continuity in measure. The construction respects the original law and its convolution roots, so no property of infinite divisibility is lost when the action is replaced by this model.

Lemma 4.2 and final proofCorrect and complete

Null Lévy base and diagonal ergodicity

Pages 10–15 · Lemma 4.2 and final proof · arXiv:2601.17988v2

If the Lévy base had a positive finite invariant part, the compound-Poisson count on that part would violate ergodicity of the process factor. Nullity of the base implies ergodicity of all finite Cartesian powers of its Poisson suspension, hence weak mixing. The Gaussian spectral criterion supplies the same conclusion for that independent factor. Products and factors of weakly mixing actions are weakly mixing, and the dense-subgroup closure argument handles the original group.

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No non-novelty findings.

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Paper: arXiv:2601.17988v2
Authors listed: Nachi Avraham-Re'em, Emmanuel Roy
Audit date: August 18, 2026

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Ergodicity implies weak mixing for separable infinitely divisible stationary processes

Poissonian and Gaussian components

Reduction through a countable dense subgroup

Theorems 3.1, Lemma 4.2, and the final proof

Stochastic realization and continuity of the action

Null Lévy base and diagonal ergodicity

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