Fourth-Moment Geometry of Rademacher Sums

2026-08-18

AI Summary

该研究解决了Rademacher和的高阶矩依赖于四阶质量的问题，通过结合固定q矩包络与低于凸性阈值的独立论证方法，确定了线性-在q界限下的Gaussian稳定性不等式。

Abstract

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs and let $a=(a_1,\ldots,a_n)\in\R^n$ satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range $p\geq4$ of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional $L_p/L_4$ Khintchine constant for $p\geq5$, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Barański, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at $p=3$. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.

Problem

• Rademacher sums
• higher moments
• fourth-order mass
• Gaussian stability inequality
• Khintchine constant

Innovation

• Gaussian stability inequality
• Khintchine constant
• Rademacher sums
• Fourth-moment