A tight Gaussian bound for weighted sums of Rademacher random variables

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 \[c_*=\bigl(4\mathbb{P}\{\eta\geq\sqrt{2}\}\bigr)^ {-1}\approx3.178.\]