A tight Gaussian bound for weighted sums of Rademacher random variables

TL;DR

This paper establishes the exact smallest constant for Gaussian bounds on weighted sums of Rademacher variables, providing a precise comparison between their tail probabilities and the standard normal distribution.

Contribution

It derives the tightest possible Gaussian bound constant for sums of Rademacher variables with bounded coefficients, improving understanding of their tail behavior.

Findings

The optimal constant c* is approximately 3.178.

The bound applies uniformly for all x in the real line.

The result sharpens previous inequalities for Rademacher sums.

Abstract

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, that is $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$. Let $S_n=a_1\varepsilon_1+\cdots+a_n\varepsilon_n$, where $\mathbf{a}=(a_1,\ldots,a_n)\in\mathbb{R}^n$ is a vector such that ${a_1^2+\cdots+a_n^2\leq1}$. We find the smallest possible constant $c$ in the inequality \[\mathbb{P}\{S_n\geq x\}\leq c\mathbb{P}\{\eta\geq x\}\qquad for all x\in \mathbb{R},\] where $\eta\sim N(0,1)$ is a standard normal random variable. This optimal value is equal to \[c_*=\bigl(4\mathbb{P}\{\eta\geq\sqrt{2}\}\bigr)^ {-1}\approx3.178.\]