A tight Gaussian bound for weighted sums of Rademacher random variables

Abstract

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent identically distributed Rademacher random variables, that is $\mathbb{P}\{{\epsilon }_i=\pm 1\}=1/2$. Let $S_n=a_1{\epsilon }_1+\cdots +a_n{\epsilon }_n$, where $\mathbf{a}=(a_1,\dots ,a_n)\in {\mathbb{R}}^n$ is a vector such that $a_1^2+\cdots +a_n^2\le 1$. We find the smallest possible constant $c$ in the inequality \[\mathbb{P}\{S_n\geq x\}\leq c\mathbb{P}\{η\geq x\}\qquad for all x\in \mathbb{R},\] where $\eta \sim N(0,1)$ is a standard normal random variable. This optimal value is equal to \[c_*=\bigl(4\mathbb{P}\{η\geq\sqrt{2}\}\bigr)^ {-1}\approx3.178.\]

Published at http://dx.doi.org/10.3150/14-BEJ603 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)