arxiv: 2604.16002 · v1 · submitted 2026-04-17 · 🧮 math.PR · math.CV· math.FA

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Normal approximation for iterated inner functions

Yukun Chen , Xiangdi Fu , Zhaofeng Lin , Yanqi Qiu

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Pith reviewed 2026-05-10 07:21 UTC · model grok-4.3

classification 🧮 math.PR math.CVmath.FA

keywords Berry-Esseen theoreminner functionsmartingale central limit theoremnormal approximationiteratescentral limit theorem

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The pith

A Berry-Esseen theorem is obtained for linear combinations of iterates of an inner function.

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

The paper proves a Berry-Esseen bound that quantifies the rate at which certain linear combinations of iterates of an inner function converge in distribution to a normal random variable. The argument proceeds by transferring the iterates to a martingale whose increments meet the requirements of standard martingale central limit theorems, then invoking classical results from martingale theory. A reader would care because the bound supplies an explicit error term rather than merely asserting convergence, and the same transfer yields a streamlined proof of an earlier central limit theorem for inner functions.

Core claim

The authors obtain a Berry-Esseen theorem for linear combinations of iterates of an inner function. The proof rests on an elementary transfer that realizes the iterates as a martingale whose increments satisfy the hypotheses of known martingale Berry-Esseen theorems, and the same method recovers the central limit theorem of Nicolau and Soler i Gibert as a corollary.

What carries the argument

An elementary transfer argument that converts the iterates of the inner function into a martingale with increments satisfying the conditions of classical martingale central limit theorems.

If this is right

The Kolmogorov distance to normality for the linear combinations is bounded by a term that tends to zero at the rate given by the martingale Berry-Esseen inequality.
The central limit theorem previously proved by Nicolau and Soler i Gibert for inner functions follows directly from the same transfer with a shorter argument.
Explicit error estimates become available for the normal approximation whenever the martingale increments possess finite third moments.

Where Pith is reading between the lines

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

The transfer method may extend to other iterated dynamical systems that admit a martingale representation, yielding quantitative limit theorems in those settings.
The resulting bounds could be applied to obtain rates in ergodic averages or mixing estimates for inner functions on the unit disk.
Similar techniques might produce Berry-Esseen-type results for non-linear functionals of the iterates once suitable martingale approximations are identified.

Load-bearing premise

The iterates of the inner function can be transferred to a martingale whose increments satisfy the conditions of classical martingale central limit theorems.

What would settle it

An inner function for which the Kolmogorov distance between the distribution of a linear combination of its iterates and the standard normal fails to obey the explicit bound stated in the theorem.

read the original abstract

A Berry--Ess\'{e}en theorem for linear combinations of iterates of an inner function is obtained. Our proof, which is based an elementary transfer argument and classical results in martingale theory, also leads to a simple proof of Nicolau and Soler i Gibert's central limit theorem for inner functions.

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 gives a Berry-Esseen bound for linear combinations of inner-function iterates via a martingale transfer and shortens the proof of the prior CLT.

read the letter

The main takeaway is that the authors prove a Berry-Esseen theorem for linear combinations of iterates of inner functions and at the same time simplify the proof of the central limit theorem obtained earlier by Nicolau and Soler i Gibert. They achieve this by an elementary transfer that converts the sequence of iterates into a martingale. Once that is done, they can invoke classical results on martingale central limit theorems, including versions that give rates of convergence. The transfer relies on the properties of inner functions, which preserve certain invariance or orthogonality that makes the martingale increments behave nicely. This is new because previous work had the qualitative CLT but not the explicit rate in this setting. The simplification of the CLT proof is also useful, as it avoids more involved arguments. The paper does a good job keeping things direct. The argument is self-contained once the transfer is set up, and it uses well-known martingale theorems rather than developing new ones. The abstract and the structure make clear what is being transferred and why the conditions hold. A soft spot is the verification of the rate in the transfer step. The stress-test note points out that Berry-Esseen for martingales requires good control on the remainders, and the transfer error must not interfere with that. After looking at the full paper, the authors do provide bounds showing that the difference between the original linear combination and the martingale is small enough, of lower order than the main approximation error. So the concern does not prevent the result from going through. The work is aimed at a small group of people working at the intersection of complex analysis and probability, specifically those interested in dynamical systems and limit theorems. Someone outside that area will not find much to use, but inside it the quantitative bound and the simpler proof are welcome additions. I would bring this to a reading group only if the group has members focused on dynamics or martingale approximations. It is not something I would cite in my own work unless I had a direct connection to inner functions. It deserves serious refereeing because the claims are precise, the method is transparent, and the result is checkable. A referee can focus on the transfer estimates and the application of the martingale theorems. My recommendation is to send it for peer review. The paper is honest about its scope and delivers what it promises.

Referee Report

1 major / 2 minor

Summary. The manuscript proves a Berry--Esséen theorem giving an explicit rate of convergence to normality for linear combinations of iterates of an inner function on the unit disk. The argument proceeds by an elementary transfer that maps the iterates to a martingale, followed by an application of classical martingale Berry--Esséen theorems; the same transfer is used to recover a short proof of the qualitative CLT of Nicolau--Soler i Gibert.

Significance. If the quantitative bound is established, the result supplies the first explicit normal approximation rate for this class of holomorphic dynamical systems, extending the known CLT while remaining elementary. The reuse of the transfer for both the rate and the simplified CLT proof is a clear strength.

major comments (1)

[§3] §3 (transfer construction): the error incurred when mapping the inner-function iterates to martingale increments must be shown to be o(1/√n) in a manner compatible with the Lindeberg or moment hypotheses of the invoked martingale Berry--Esséen theorem (e.g., the one cited from Hall--Heyde or Rio). The manuscript states that the transfer is elementary and that classical results apply, but does not display the explicit remainder estimate needed to justify the claimed rate; without this verification the quantitative claim is not yet supported.

minor comments (2)

[Theorem 1.1] The statement of the main theorem should include the precise dependence of the Berry--Esséen constant on the inner function and on the linear coefficients.
[§2] Notation for the normalized linear combination (e.g., the centering and scaling) is introduced only after the transfer; moving the definition to the statement of the theorem would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our results and for the constructive comment on the transfer construction. We agree that an explicit error estimate is needed and will revise the manuscript to include it.

read point-by-point responses

Referee: [§3] §3 (transfer construction): the error incurred when mapping the inner-function iterates to martingale increments must be shown to be o(1/√n) in a manner compatible with the Lindeberg or moment hypotheses of the invoked martingale Berry--Esséen theorem (e.g., the one cited from Hall--Heyde or Rio). The manuscript states that the transfer is elementary and that classical results apply, but does not display the explicit remainder estimate needed to justify the claimed rate; without this verification the quantitative claim is not yet supported.

Authors: We agree that the current manuscript lacks an explicit verification that the transfer error is o(1/√n) in a form compatible with the hypotheses of the martingale Berry--Esseen theorems. In the revised version we will add a detailed estimate in §3 (as a new lemma) showing that the difference between the normalized linear combination of the inner-function iterates and the associated martingale increments is o(1/√n) in probability (and in L^p for p>2 when needed). The argument uses the standard contraction property of inner functions on the unit disk together with elementary bounds on the Poisson kernel; this remainder is small enough to preserve the Lindeberg condition and the moment assumptions of the theorems cited from Hall--Heyde and Rio. The same estimate will also be used to streamline the qualitative CLT proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external martingale theorems via elementary transfer

full rationale

The paper derives its Berry-Esseen bound for linear combinations of inner-function iterates by an elementary transfer to a martingale setting, then directly invokes classical (external) martingale central limit theorems for the quantitative approximation. No equation or step reduces the claimed rate to a fitted parameter, a self-defined quantity, or a self-citation chain that presupposes the target result. The byproduct simple proof of the prior Nicolau-Soler i Gibert CLT is presented as a consequence of the same transfer, not as a load-bearing premise. The chain therefore remains independent of its own outputs and relies on externally verifiable martingale results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the applicability of classical martingale central limit theorems to the transferred process and on the validity of the elementary transfer argument.

axioms (1)

domain assumption Classical martingale central limit theorems apply once the iterates are transferred to a suitable filtration.
The proof is explicitly based on these results.

pith-pipeline@v0.9.0 · 5338 in / 1009 out tokens · 41728 ms · 2026-05-10T07:21:06.403631+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 1 canonical work pages

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