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Khinchin inequalities for uniforms on spheres with a deficit

Jacek Jakimiuk, Colin Tang, Tomasz Tkocz

TL;DR

This work extends sharp Khinchin-type moment comparison inequalities from Rademacher sums to sums of independent vectors uniformly distributed on Euclidean spheres, by introducing an optimal deficit term that quantifies the gap to a Gaussian benchmark. The authors develop a Lindeberg-style swapping framework and a quantitative convexity analysis of an auxiliary functional, yielding explicit deficit bounds that scale as $c_{p,d}\sum a_j^4$ (and a stability deficit involving $\sum(\frac{1}{n}-a_j^2)^2$). The results provide both non-asymptotic deficit bounds and asymptotic sharpness ($c_{p,d}=\Theta_p(1/d)$ as $d\to\infty$), with a diagonalized, stability-focused variant. Overall, the paper advances the understanding of stability in high-dimensional Khinchin inequalities and introduces tools that connect convexity, Lindeberg arguments, and spherical symmetry to precise deficit estimates, with potential implications for related moment comparison problems in high dimensions.

Abstract

We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).

Khinchin inequalities for uniforms on spheres with a deficit

TL;DR

This work extends sharp Khinchin-type moment comparison inequalities from Rademacher sums to sums of independent vectors uniformly distributed on Euclidean spheres, by introducing an optimal deficit term that quantifies the gap to a Gaussian benchmark. The authors develop a Lindeberg-style swapping framework and a quantitative convexity analysis of an auxiliary functional, yielding explicit deficit bounds that scale as (and a stability deficit involving ). The results provide both non-asymptotic deficit bounds and asymptotic sharpness ( as ), with a diagonalized, stability-focused variant. Overall, the paper advances the understanding of stability in high-dimensional Khinchin inequalities and introduces tools that connect convexity, Lindeberg arguments, and spherical symmetry to precise deficit estimates, with potential implications for related moment comparison problems in high dimensions.

Abstract

We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).

Paper Structure

This paper contains 9 sections, 6 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. New results
  3. Previous work
  4. Proofs
  5. Overview
  6. Main lemmas
  7. An auxiliary lemma
  8. Proof of Theorem \ref{['thm:main']}
  9. Proof of Theorem \ref{['thm:main-diag']}

Key Result

Theorem 1

Let $d \geq 2$ be a fixed dimension, let $\xi_1, \xi_2, \dots$ be independent random vectors uniform on the unit Euclidean sphere $S^{d-1}$ in $\mathbb{R}^d$ and let $Z$ be a Gaussian random vector in $\mathbb{R}^d$ with mean $0$ and covariance $\frac{1}{d}I_d$. Let $p \geq 2$. For every $n \geq 1$, where

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 2 more