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May 2015 A tight Gaussian bound for weighted sums of Rademacher random variables
Vidmantas Kastytis Bentkus, Dainius Dzindzalieta
Bernoulli 21(2): 1231-1237 (May 2015). DOI: 10.3150/14-BEJ603

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

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

Citation

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Vidmantas Kastytis Bentkus. Dainius Dzindzalieta. "A tight Gaussian bound for weighted sums of Rademacher random variables." Bernoulli 21 (2) 1231 - 1237, May 2015. https://doi.org/10.3150/14-BEJ603

Information

Published: May 2015
First available in Project Euclid: 21 April 2015

zbMATH: 1332.60064
MathSciNet: MR3338662
Digital Object Identifier: 10.3150/14-BEJ603

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

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Vol.21 • No. 2 • May 2015
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