The Annals of Statistics

A Probability Inequality for Linear Combinations of Bounded Random Variables

Morris L. Eaton

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Abstract

Let $Y_1, \cdots, Y_n$ be independent random variables with mean zero such that $|Y_i| \leqq i, i = 1, \cdots, n$, and let $\theta_1,\cdots, \theta_n$ be real numbers satisfying $\sum^n_1 \theta_i^2 = 1$. Set $S_n(\theta) = \sum^n_1 \theta_i Y_i$ and let $\varphi(x) = (2\pi)^{-\frac{1}{2}} \exp \lbrack - \frac{1}{2} x^2\rbrack$. THEOREM. For $\alpha > 0$, and for all $\theta_1,\cdots, \theta_n$, \begin{\align*}P\{|S_n(\theta)| \geqq \alpha\} &\leqq 2\inf_{0\leqq u \leqq \alpha} \int^\infty_u \frac{(x - u)^3}{(\alpha - u)^3} \varphi (x) dx \\ &\leqq 12 \frac{\varphi(\alpha)}{\alpha} \inf_{0\leqq \delta \leqq \alpha^2} \frac{\exp\lbrack\delta/2(2 - \delta/\alpha^2)\rbrack}{\delta^3(1 - \delta/\alpha^2)^4}.\\ \end{align*}

Article information

Source
Ann. Statist., Volume 2, Number 3 (1974), 609-614.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342725

Digital Object Identifier
doi:10.1214/aos/1176342725

Zentralblatt MATH identifier
0282.62012

JSTOR
links.jstor.org

Keywords
6210 Probability inequality bounded random variables sums

Citation

Eaton, Morris L. A Probability Inequality for Linear Combinations of Bounded Random Variables. Ann. Statist. 2 (1974), no. 3, 609--614. doi:10.1214/aos/1176342725. https://projecteuclid.org/euclid.aos/1176342725


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