Bernoulli

  • Bernoulli
  • Volume 21, Number 2 (2015), 1231-1237.

A tight Gaussian bound for weighted sums of Rademacher random variables

Vidmantas Kastytis Bentkus and Dainius Dzindzalieta

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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

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

Article information

Source
Bernoulli, Volume 21, Number 2 (2015), 1231-1237.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1429624976

Digital Object Identifier
doi:10.3150/14-BEJ603

Mathematical Reviews number (MathSciNet)
MR3338662

Zentralblatt MATH identifier
1332.60064

Keywords
bounds for tail probabilities Gaussian large deviations optimal constants random sign self-normalized sums Student’s statistic symmetric tail comparison weighted Rademachers

Citation

Bentkus, Vidmantas Kastytis; Dzindzalieta, Dainius. A tight Gaussian bound for weighted sums of Rademacher random variables. Bernoulli 21 (2015), no. 2, 1231--1237. doi:10.3150/14-BEJ603. https://projecteuclid.org/euclid.bj/1429624976


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