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arxiv: 2210.07204 · v5 · submitted 2022-10-13 · 🧮 math.PR · math.DS

Non-uniform Berry-Esseen theorems for weakly dependent random variables

Yeor Hafouta This is my paper

Pith reviewed 2026-05-24 10:22 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords non-uniform Berry-Esseenweakly dependent sequencesMarkov chainsdynamical systemsrandom matriceslocal statisticscentral limit theoremconvergence rates
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The pith

Non-uniform Berry-Esseen estimates hold for several classes of weakly dependent random variable sequences.

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

This paper derives non-uniform Berry-Esseen type estimates for the rate at which sums of weakly dependent random variables approach a normal distribution. The estimates apply to uniformly elliptic inhomogeneous Markov chains, random and time-varying partially hyperbolic or expanding dynamical systems, products of random matrices, and certain local statistics. A reader might care because these bounds vary with the point in the distribution, potentially giving tighter control than uniform versions for applications involving dependent data. The results cover multiple structures that exhibit weak dependence but are not independent.

Core claim

The central claim is that non-uniform Berry-Esseen type estimates can be obtained for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics.

What carries the argument

Non-uniform Berry-Esseen bounds adapted to weak dependence conditions such as uniform ellipticity and hyperbolicity.

If this is right

  • Sums of variables from uniformly elliptic inhomogeneous Markov chains satisfy non-uniform normal approximation bounds.
  • Iterates of random and time-varying partially hyperbolic dynamical systems admit non-uniform Berry-Esseen estimates.
  • Products of random matrices yield non-uniform rates of convergence to normality for associated statistics.
  • Certain classes of local statistics on these structures also obey the non-uniform error bounds.

Where Pith is reading between the lines

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

  • The non-uniform bounds may sharpen tail probability estimates compared with uniform Berry-Esseen results in the same classes.
  • These estimates could support refined error analysis in ergodic averages arising from the listed dynamical systems.
  • The method might extend to other weak dependence notions if the core contraction or ellipticity arguments can be verified.

Load-bearing premise

The sequences must belong to one of the listed classes and satisfy the corresponding weak dependence properties such as uniform ellipticity or hyperbolicity.

What would settle it

A concrete calculation for a specific uniformly elliptic inhomogeneous Markov chain where the pointwise approximation error exceeds the non-uniform bound given by the theorem.

read the original abstract

We obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics.

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.

Referee Report

1 major / 0 minor

Summary. The paper claims to obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices, and some classes of local statistics.

Significance. If the stated non-uniform bounds hold under the respective weak-dependence conditions (uniform ellipticity, hyperbolicity, etc.), the work would extend classical Berry-Esseen results to non-uniform settings for dependent processes, which is of interest in probability theory for applications involving rates of convergence in the CLT. The abstract presents the results as derived directly from the dependence properties of the listed classes, with no evident circularity.

major comments (1)
  1. No theorems, conditions, proofs, or derivations are accessible from the provided material (only the abstract is given). This prevents any verification that the mathematics supports the claimed non-uniform estimates or that the dependence properties imply the stated rates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The referee appears to have received only the abstract, as the full manuscript with all theorems, conditions, proofs, and derivations is available on arXiv:2210.07204. We address the comment below.

read point-by-point responses

  1. Referee: No theorems, conditions, proofs, or derivations are accessible from the provided material (only the abstract is given). This prevents any verification that the mathematics supports the claimed non-uniform estimates or that the dependence properties imply the stated rates.

    Authors: The complete manuscript, including all stated theorems, conditions on weak dependence (uniform ellipticity, hyperbolicity, etc.), proofs, and derivations, is publicly available on arXiv:2210.07204. The results are derived directly from the respective dependence properties without circularity, as described in the abstract. If the referee was provided only the abstract, we are happy to supply the full text or specific sections for review. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents non-uniform Berry-Esseen bounds as consequences of the weak dependence properties (uniform ellipticity, hyperbolicity, etc.) satisfied by the listed classes of sequences. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation chain relies on standard probabilistic estimates applied to the given dependence conditions without tautological reduction. The abstract and structure indicate an independent derivation from the model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the work rests on standard probability axioms and domain-specific definitions of weak dependence for the listed systems, with no free parameters, invented entities, or ad-hoc constructions mentioned.

axioms (2)
  • standard math Standard axioms of probability theory and measure theory
    Required to define random variables, expectations, distributions, and convergence in distribution.
  • domain assumption Weak dependence conditions specific to each class (uniform ellipticity, hyperbolicity, etc.)
    The estimates are stated to hold under these properties of Markov chains, dynamical systems, and matrix products.

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Reference graph

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