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A Sharp Gaussian Tail Bound for Sums of Uniforms

Xinjie He, Tomasz Tkocz, Katarzyna Wyczesany

Abstract

We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.

A Sharp Gaussian Tail Bound for Sums of Uniforms

Abstract

We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.
Paper Structure (11 sections, 5 theorems, 46 equations, 1 table)

This paper contains 11 sections, 5 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proofs
  3. Overview
  4. Approach for small $t$.
  5. Approach for large $t$.
  6. Proof of Theorem \ref{['thm:main']}
  7. Proofs of the auxiliary lemmas
  8. Proof of Lemma \ref{['lm:BK']} (sketch)
  9. Proof of Lemma \ref{['lm:G']}
  10. Proof of Lemma \ref{['lm:ave-tail']}
  11. Final remarks

Key Result

Theorem 1

Let $U_1, U_2, \dots$ be independent random variables uniform on $[-1,1]$. For every $n \geq 1$, real numbers $a_1, \dots, a_n$ with $\sum_{j=1}^n a_j^2 = 1$ and positive $t$, we have where $G$ is a standard Gaussian random variable (mean $0$, variance $1$) and the constant equals (the supremum attained uniquely at $t_0 = 0.642908..$).

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2: Barthe-Koldobsky
  • Lemma 3: Barthe-Koldobsky
  • Lemma 4
  • Lemma 5
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 1 more