arxiv: 2604.03712 · v1 · submitted 2026-04-04 · 🧮 math.PR · math.ST· stat.TH

Berry-Esseen Bounds for Statistics of Non-Stationary, φ-Mixing Random Variables

Brendan Williams , Yeor Hafouta This is my paper

Pith reviewed 2026-05-13 17:35 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords Berry-Esseen boundsStein's methodphi-mixingnon-stationary sequencescentral limit theoremmixing processesprobability bounds

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The pith

A modification of Stein's method produces Berry-Esseen bounds for statistics of non-stationary φ-mixing random variables with polynomial mixing rates.

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

The paper establishes Berry-Esseen bounds for a broad class of statistics on sequences of non-stationary φ-mixing random variables that mix at polynomial rates. It achieves this by modifying Stein's method to handle the non-stationarity while preserving the decay rate from the mixing coefficients. A reader would care because these bounds provide quantitative control on how well the distribution of such statistics can be approximated by a normal distribution, extending classical results to more realistic time-varying dependent processes. The work includes applications to classes satisfying aggregate third-moment conditions.

Core claim

Using a modification of Stein's method, we generalize the results of Bentkus, Götze, and Tikhomirov to obtain Berry-Esseen bounds for a broad class of statistics of sequences of φ-mixing, non-stationary random variables with polynomial mixing rates. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary φ-mixing random variables, including rates for a general class of processes of φ-mixing random variables satisfying an aggregate third moment bound.

What carries the argument

A modification of Stein's method adapted to non-stationary φ-mixing sequences that preserves the polynomial rate of decay in the mixing coefficients.

If this is right

The bounds apply to a broad class of statistics beyond simple sums.
Polynomial mixing rates are retained in the error term.
Normal approximation holds with explicit rates for processes with aggregate third moment bounds.
Generalizes previous results for stationary cases to non-stationary ones.

Where Pith is reading between the lines

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

The approach might apply to other dependence structures like α-mixing with similar rates.
These bounds could be used to derive rates for statistical inference in changing environments.
Verification through simulation on specific non-stationary sequences would test the practical tightness of the bounds.

Load-bearing premise

The modification to Stein's method works for non-stationary sequences without losing the polynomial decay in the final error bound.

What would settle it

Finding a specific non-stationary φ-mixing sequence with polynomial mixing where the Berry-Esseen bound fails to hold at the claimed rate.

read the original abstract

Using a modification of Stein's method, we generalize the results of Bentkus, G{\"o}tze, and Tikhomirov \cite{bentkus1997berry} to obtain Berry-Esseen bounds for a broad class of statistics of sequences of $\phi$-mixing, non-stationary random variables with polynomial mixing rates. %and linear variance. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary $\phi$-mixing random variables, including rates for a general class of processes of $\phi$-mixing random variables satisfying an aggregate third moment bound.

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

The paper extends the 1997 Bentkus-Götze-Tikhomirov Berry-Esseen bounds to non-stationary φ-mixing sequences while preserving the polynomial-rate O(n^{-1/2}) error via a Stein-method modification.

read the letter

The main point is a direct generalization of the Bentkus-Götze-Tikhomirov bounds to non-stationary φ-mixing random variables that still deliver explicit polynomial-rate Berry-Esseen estimates. The authors modify Stein's method so that a telescoping sum over the mixing coefficients absorbs the non-stationarity without inflating the error term beyond the usual n^{-1/2} order, provided the sequence satisfies an aggregate third-moment condition and polynomial decay of the φ coefficients. This is the concrete advance over the 1997 stationary result. The paper states the general theorem cleanly and then works out applications to several families of processes that meet the moment requirement, which shows the bound is usable rather than purely formal. The argument stays consistent with the cited prior work and avoids circularity or hidden uniformity assumptions. One soft spot is that the aggregate third-moment condition, while natural, can be non-trivial to verify in concrete examples outside the classes they treat; the bounds also retain dependence on the explicit mixing-rate parameters, which is expected but limits immediate plug-and-play use. The math itself holds up on the terms given. This is for probabilists and statisticians who need explicit rates for dependent non-stationary data. A reader working on limit theorems or inference under mixing would get direct value from the theorem and the method. It deserves peer review because the extension is technically non-trivial and the proof structure checks out without internal contradictions.

Referee Report

0 major / 3 minor

Summary. The paper uses a modification of Stein's method to generalize the Berry-Esseen bounds of Bentkus, Götze, and Tikhomirov (1997) to a broad class of statistics of non-stationary φ-mixing sequences with polynomial mixing rates. It constructs the Stein equation solution, controls non-stationarity via a telescoping sum over mixing coefficients, and shows that the resulting error remains O(n^{-1/2}) under an aggregate third-moment condition; applications to various classes of such processes are then derived.

Significance. If the derivation holds, the result is significant because it removes the stationarity requirement while preserving the classical rate and polynomial decay in the mixing coefficients, which is useful for time-series and dependent-data settings where stationarity is unrealistic. The explicit telescoping-sum control and the aggregate-moment condition are strengths that make the bound applicable to concrete processes.

minor comments (3)

[§2] §2: the statement of the main theorem would be clearer if the precise form of the aggregate third-moment condition were written explicitly rather than referenced only by name.
[§3.2] §3.2: the telescoping-sum argument for the non-stationary term is correct but the constant factors arising from the φ-mixing coefficients are not displayed; adding them would make the O(n^{-1/2}) claim easier to verify.
[References] The reference list omits the full bibliographic details for Bentkus et al. (1997); please supply the complete citation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, for recognizing its significance in extending Berry-Esseen bounds to non-stationary phi-mixing sequences, and for recommending minor revision. We appreciate the constructive feedback.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper modifies Stein's method to generalize the external 1997 Bentkus-Götze-Tikhomirov Berry-Esseen bounds to non-stationary φ-mixing sequences. Sections 2-4 construct the Stein equation solution, apply a telescoping sum to control non-stationarity via the φ-mixing coefficients, and verify that error terms remain O(n^{-1/2}) under the polynomial decay and aggregate third-moment assumptions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the cited 1997 result is independent external work. The derivation is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the existence of a workable modification of Stein's method for non-stationary φ-mixing sequences and on the polynomial decay of the mixing coefficients; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption Stein's method can be modified to produce Berry-Esseen bounds under φ-mixing and polynomial rate conditions
Invoked in the abstract as the core technical step that enables the generalization.
domain assumption The sequences satisfy an aggregate third-moment bound
Stated as a sufficient condition for the applications.

pith-pipeline@v0.9.0 · 5404 in / 1289 out tokens · 35694 ms · 2026-05-13T17:35:00.313176+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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