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Moment defects and sharp iid Berry–Esseen bounds

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitycs.IT — Information Theory

Contributed by Kacper Rodziewicz, Bartosz Kołodziejek ↗

Version 2 / Oct 07, 2026 / CC BY 4.0

Abstract

Starting from Shevtsova’s sharp third-moment inequality, we develop an exact defect from the standardized two-point family and derive quantitative transport and characteristic-function bounds. A penalized extremal problem yields eventual two-point optimality, Cn=CQ∗(n)<CEC_n=C_Q^*(n)<C_{\mathrm E}, together with a uniform linear deficit and sharp cubic Wasserstein stability. A fifth-order lattice saddle-point expansion gives the full oscillatory n−2n^{-2} correction to the optimal constants. A separate effective argument proves Cn<CEC_n<C_{\mathrm E} for every n≥31n\ge31, using finite-index certificates and a first-failure argument. Exact polynomial certificates give unrestricted bounds for two and three summands. For four summands, a common-measure Fourier comparison and rational certificates yield the unrestricted bound C4≤0.4097<CEC_4\le0.4097<C_{\mathrm E}; on laws supported on at most three points, we obtain the sharper bound C4,≤3<0.409C_{4,\le3}<0.409. An opening analytic argument explains the passage from Rademacher comparison to the matching two-point law. An elementary historical proof of C≤0.45C\le0.45 is included in 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

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Version history

  1. v1Submitted by Kacper RodziewiczInitial depositSupersededOct 07, 2026
  2. v2Submitted by Kacper RodziewiczSubmitted as a new versionCurrentOct 12, 2026