math.PR — Probability

Moment defects and sharp iid Berry–Esseen bounds

Contributed by Kacper Rodziewicz, Bartosz Kołodziejek

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, $C_n=C_Q^*(n)<C_{\mathrm E}$, together with a uniform linear deficit and sharp cubic Wasserstein stability. A fifth-order lattice saddle-point expansion gives the full oscillatory $n^{-2}$ correction to the optimal constants. A separate effective argument proves $C_n<C_{\mathrm E}$ for every $n\ge31$, 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 $C_4\le0.4097<C_{\mathrm E}$; on laws supported on at most three points, we obtain the sharper bound $C_{4,\le3}<0.409$. An opening analytic argument explains the passage from Rademacher comparison to the matching two-point law. An elementary historical proof of $C\le0.45$ 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.