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

\[\mathbb{P}\{S_{n}\geq x\}\leq c\mathbb{P}\{\eta\geq x\}\qquad\mbox{for all }x\in \mathbb{R},\] where $\eta\sim N(0,1)$ is a standard normal random variable. This optimal value is equal to

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