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Moment defects and sharp iid Berry–Esseen bounds

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Kacper Rodziewicz ↗, Bartosz Kołodziejek ↗

Version 1 / Oct 07, 2026 / CC BY 4.0

Abstract

The known sharp third-moment inequality supplies an exact defect that identifies the standardized two-point family and controls transport and characteristic functions. We develop the corresponding Fourier reduction and give the qualitative eventual upper-bound argument. A penalized extremal problem yields eventual two-point optimality and a uniform linear deficit, with sharp cubic Wasserstein stability. A fifth-order lattice saddle gives the full oscillatory n−2n^{-2} correction for the optimal constants. A separate effective argument proves Cn<CEC_n<C_{\mathrm E} for every n≥49n\ge49, by a finite base and a first-failure induction. Exact polynomial certificates give bounds for two and three summands; a complete interval argument treats all three-point laws at four summands. An elementary historical bound C≤0.45C\le0.45 is included as an appendix. ChatGPT and Codex assisted the development of the proofs and computational certificates and the preparation of this manuscript, using the GPT-5.6 Sol, GPT-6.1 Sol and GPT-6 Astra models.

Provenance statement

This work was developed with substantive assistance from ChatGPT and Codex, using GPT-5.6 Sol, GPT-6.1 Sol, and GPT-6 Astra. These tools supported mathematical exploration, the development and revision of proofs, literature searches and comparisons, computational certificate construction, and manuscript preparation. Parallel AI-agent roles were used to develop arguments, implement calculations, and critically examine proposed proofs. The computational arguments use rational polynomial and Bernstein certificates and outward-rounded interval arithmetic. Lean 4 and mathlib were used for partial, conditional formalization; the entire manuscript is not claimed to have been formally verified. AI-generated arguments were treated as proposals requiring mathematical verification, not as independent evidence of correctness. The named human authors remain responsible for the mathematical claims, attribution of prior work, and final manuscript.

References

  1. V. Bentkus and K. Kirša, Estimates of the closeness of a distribution to the normal law, Lithuanian Mathematical Journal 29 (1989), 321--332.
  2. L. Mattner and I. Shevtsova, An optimal Berry--Esseen type theorem for integrals of smooth functions, ALEA 16 (2019), 487--530. r̆l https://doi.org/10.30757/ALEA.v16-19 .DOI
  3. I. Shevtsova, On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications, Journal of Mathematical Analysis and Applications 418 (1) (2014), 185--210. r̆l https://doi.org/10.1016/j.jmaa.2014.03.075 .DOI
  4. I. Shevtsova, Moment-type estimates with asymptotically optimal structure for the accuracy of the normal approximation, Annales Mathematicae et Informaticae 39 (2012), 241--307. r̆l https://ami.uni-eszterhazy.hu/uploads/papers/finalpdf/AMI_39_from241to307.pdf .link
  5. H. Prawitz, Limits for a distribution, if the characteristic function is given in a finite domain, Skandinavisk Aktuarietidskrift (1972), 138--154. r̆l https://doi.org/10.1080/03461238.1972.10404645 .DOI
  6. J. Schulz, The optimal Berry--Esseen constant in the binomial case , doctoral dissertation, Universität Trier, 2016. r̆l https://d-nb.info/1197702695/34 .link
  7. I. G. Shevtsova, On the absolute constants in the Berry--Esseen inequality and its structural and nonuniform improvements, Informatika i Ee Primeneniya 7 (1) (2013), 124--125. r̆l https://www.mathnet.ru/eng/ia252 .link
  8. H. He and G. Cheng, The Berry--Esseen Bound is Sharp for All Sufficiently Large Sample Sizes , arXiv:2609.06358v1 (2026). r̆l https://arxiv.org/abs/2609.06358v1 .arXiv
  9. A. Zolotukhin, S. Nagaev and V. Chebotarev, On a bound of the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables, Modern Stochastics: Theory and Applications 5 (2018), 385--410.
  10. K. Rodziewicz, An elementary interval certificate for the Berry--Esseen bound 0.450.45 , certificate archive (2026). r̆l https://github.com/rodziewiczk/berry-esseen-045-certificate .link

Version history

  1. v1Submitted by Kacper RodziewiczInitial depositCurrentOct 07, 2026