A Probability Inequality for Linear Combinations of Bounded Random Variables

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*}