Almost sure central limit theorems via chaos expansions and related results

Guangqu Zheng; Leonardo Maini; Maurizia Rossi

arxiv: 2502.00759 · v3 · pith:5GWZTTKInew · submitted 2025-02-02 · 🧮 math.PR

Almost sure central limit theorems via chaos expansions and related results

Leonardo Maini , Maurizia Rossi , Guangqu Zheng This is my paper

Pith reviewed 2026-05-23 04:30 UTC · model grok-4.3

classification 🧮 math.PR

keywords almost sure central limit theoremWiener chaos expansionMalliavin-Stein methodstationary Gaussian fieldsBerry's random wave modelBessel functionsBreuer-Major theoremexcursion volume

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The pith

Stationary Gaussian random fields satisfy almost sure central limit theorems for integral functionals under mild covariance conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an almost sure central limit theorem for integral functionals of stationary Gaussian random fields as the integration domain expands to all of space. This holds via Wiener chaos expansions and the Malliavin-Stein method applied only under mild conditions on the covariance function, without any prior assumption of Malliavin differentiability or other regularity. The same analysis produces a quantitative central limit theorem with explicit Wasserstein convergence rates, proves regularity properties of the functionals, and confirms Malliavin differentiability of the excursion volume for Berry's random wave model. It also delivers the precise asymptotic rate, in terms of the exponent q, for the q-th moment of Bessel functions. The results are applied directly to Breuer-Major theorems and to Berry's model.

Core claim

Using the Wiener chaos expansion and Malliavin-Stein method, the paper proves that integral functionals of stationary Gaussian random fields obey an almost sure central limit theorem whenever the covariance satisfies mild conditions that permit the expansion. The argument simultaneously yields a quantitative central limit theorem in quadratic Wasserstein distance, establishes certain regularity properties, resolves the open question of Malliavin differentiability for the excursion volume of Berry's random wave model, and gives the exact asymptotic rate for the q-th moment of Bessel functions as a function of q. These conclusions apply to Breuer-Major central limit theorems and to Berry's own

What carries the argument

Wiener chaos expansion combined with the Malliavin-Stein method applied to integral functionals of stationary Gaussian random fields.

If this is right

The excursion volume of Berry's random wave model is Malliavin differentiable.
The q-th moments of Bessel functions admit an exact asymptotic rate that depends on q and confirms prior numerical conjectures.
Almost sure central limit theorems hold for Breuer-Major central limit theorems under the same mild covariance conditions.
Almost sure central limit theorems hold for Berry's random wave model.
Quantitative rates of convergence in quadratic Wasserstein distance are available for the central limit theorems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method supplies a template for proving almost sure central limit theorems in other Gaussian settings where direct verification of differentiability is difficult.
The exact Bessel moment asymptotics supply a precise benchmark that can be checked against independent numerical or analytic approximations of the same integrals.

Load-bearing premise

The covariance function of the stationary Gaussian fields meets mild conditions that let the Wiener chaos expansion and Malliavin-Stein method deliver the almost sure central limit theorem without any a-priori Malliavin differentiability.

What would settle it

A concrete stationary Gaussian field whose covariance satisfies the stated mild conditions yet whose integral functional fails to satisfy the almost sure central limit theorem, or direct computation showing that the q-th moment of the Bessel function deviates from the predicted exact asymptotic rate.

read the original abstract

In this work, we investigate the asymptotic behavior of integral functionals of stationary Gaussian random fields as the integration domain tends to be the whole space. More precisely, using the Wiener chaos expansion and Malliavin-Stein method, we establish an {\it almost sure central limit theorem} (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian fields. In this setting, we additionally derive a {\it quantitative central limit theorem} with rate of convergence in quadratic Wasserstein distance, and show certain regularity property for the said integral functionals. In particular, we solve an open question on the {\it Malliavin differentiability of the excursion volume of Berry's random wave model}. As a key consequence of our analysis, we obtain the exact asymptotic rate (as a function of the exponent $q$) for the $q$-th moment of Bessel functions, thus confirming a conjecture based on existing numerical simulations. In the end, we provide two applications of our result: (i) ASCLT in the context of Breuer-Major central limit theorems, (ii) ASCLT for Berry's random wave model. Our approach does not require any knowledge on the regularity properties of random variables (e.g., Malliavin differentiability) and hence not only complements the existing literature, but also leads to novel results that are of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper derives ASCLTs for Gaussian field integrals under mild covariance decay via chaos methods, then uses the setup to prove Malliavin differentiability for Berry wave excursions and pin down the exact Bessel q-moment rate.

read the letter

The main contribution is an almost sure CLT for integral functionals of stationary Gaussian fields over expanding domains, obtained from Wiener chaos expansions plus Malliavin-Stein bounds, but only under mild covariance conditions. They also extract a quantitative CLT in Wasserstein distance and some regularity statements. The standout parts are the application to the excursion volume of Berry's random wave model, which settles the open question of its Malliavin differentiability, and the exact asymptotic rate (in q) for the q-th moments of Bessel functions, matching earlier numerical guesses. They close with applications to Breuer-Major CLTs and the random wave model itself. The claim that the method needs no prior regularity knowledge is the selling point and lets them reach the differentiability result as a consequence rather than an assumption. That looks like genuine progress if the proofs hold. The covariance conditions appear standard for these techniques, and the abstract gives no sign of circular reasoning in the citation pattern or setup. A soft spot is whether the mild decay really places the indicator-based excursion functional inside the chaos space needed for the expansion and Stein bound to close without some hidden regularity that would have made the open question easier to begin with. The stress-test note flags exactly this possibility, and the full argument would have to show how they avoid it, perhaps via approximation or direct verification. The Bessel rate is presented as a byproduct, so its strength rides on the same expansion. This is aimed at specialists in stochastic analysis and Gaussian fields who care about almost sure limits and Malliavin regularity. A reader working on related limit theorems or random waves would find the specific results and the method useful. It deserves a serious referee because it targets concrete open questions with standard tools applied in a new way.

Referee Report

1 major / 1 minor

Summary. The manuscript uses Wiener chaos expansions and the Malliavin-Stein method to prove almost sure central limit theorems (ASCLT) for integral functionals of stationary Gaussian random fields under mild covariance conditions. It also derives quantitative CLTs in quadratic Wasserstein distance, establishes regularity properties (including Malliavin differentiability of the excursion volume for Berry's random wave model), obtains exact asymptotic rates for the q-th moments of Bessel functions (confirming a numerical conjecture), and gives applications to Breuer-Major theorems and Berry's model. The approach is presented as not requiring a priori knowledge of regularity properties such as Malliavin differentiability.

Significance. If the central claims hold without gaps, the work would be significant: it resolves an open question on Malliavin differentiability of excursion volumes, supplies ASCLTs under weaker assumptions than typical in the literature, and confirms a conjecture on Bessel moments via exact rates. The byproduct results on moments and the applications to Breuer-Major and random waves are of independent interest.

major comments (1)

[Abstract and §1] Abstract and §1: the claim that ASCLT and chaos expansions hold under only mild covariance conditions without any Malliavin differentiability knowledge is load-bearing for the main theorems, yet the skeptic's note identifies a potential circularity—if the indicator functional for excursion volume is not automatically in the requisite chaos space under the stated covariance decay, the expansion and Stein bound may implicitly require the differentiability the paper claims to prove. A concrete verification that the mild conditions suffice to justify the expansion a priori is needed.

minor comments (1)

The precise statement of the 'mild conditions' on the covariance function should be stated explicitly in the introduction rather than left to the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for identifying a potential circularity concern in our claims about chaos expansions under mild covariance conditions. We address this point directly below and propose a targeted clarification.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1: the claim that ASCLT and chaos expansions hold under only mild covariance conditions without any Malliavin differentiability knowledge is load-bearing for the main theorems, yet the skeptic's note identifies a potential circularity—if the indicator functional for excursion volume is not automatically in the requisite chaos space under the stated covariance decay, the expansion and Stein bound may implicitly require the differentiability the paper claims to prove. A concrete verification that the mild conditions suffice to justify the expansion a priori is needed.

Authors: The Wiener chaos expansion applies to any square-integrable functional of the underlying Gaussian field; membership in L² is guaranteed directly by the mild covariance decay assumptions (via explicit variance computations or Breuer-Major-type integrability), without invoking Malliavin differentiability. The Malliavin-Stein bounds are then derived from the chaos expansion coefficients. Malliavin differentiability of the excursion volume is established independently in a later section as a separate result, not as a prerequisite for the expansion itself. Thus the argument is not circular. We will insert a short clarifying paragraph at the end of §1 that explicitly separates the L² justification of the expansion from the subsequent differentiability proof. revision: partial

Circularity Check

0 steps flagged

No circularity; established Wiener chaos and Malliavin-Stein methods applied to new settings without reduction to inputs

full rationale

The paper applies Wiener chaos expansion and Malliavin-Stein method to integral functionals of stationary Gaussian fields under mild covariance conditions. It explicitly states the approach requires no prior knowledge of regularity properties such as Malliavin differentiability, and derives such regularity (including for Berry's random wave model excursion volume) as a consequence. No self-citations, fitted parameters renamed as predictions, ansatzes smuggled via citation, or self-definitional steps are present in the abstract or description. The central claims rest on external established techniques rather than reducing to the paper's own inputs by construction. This is the normal non-finding for papers that extend known methods without circular load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard tools from stochastic analysis; no free parameters, invented entities, or ad hoc axioms apparent from abstract.

axioms (2)

standard math Wiener chaos expansions apply to stationary Gaussian fields
Established framework in probability theory
standard math Malliavin-Stein method yields quantitative CLTs
Standard technique for rates in central limit theorems

pith-pipeline@v0.9.0 · 5775 in / 1218 out tokens · 60168 ms · 2026-05-23T04:30:25.741544+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

using the Wiener chaos expansion and Malliavin-Stein method, we establish an almost sure central limit theorem (ASCLT) only under mild conditions on the covariance function
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we solve an open question on the Malliavin differentiability of the excursion volume of Berry's random wave model

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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