The Annals of Mathematical Statistics

On the Probability of Large Deviations for Sums of Bounded Chance Variables

Harry Weingarten

Full-text: Open access

Abstract

The following theorems are proved. THEOREM 1. If $x_1, x_2, \cdots$ satisfy $-1 \leqq x_n \leqq a, a \leqq 1$ and $E(x_n \mid x_1, \cdots, x_{n-1}) \leqq - u \max (| x_n | \mid x_1, \cdots, x_{n-1}), 0 < u < 1$, then for any positive $t$ $$\mathrm{Pr}\{x_1 + \cdots + x_n \geqq t \text{for some} n\} \leqq \theta^t,$$ where $\theta$ is the positive root (other than $\theta = 1$) of \begin{equation*}\tag{1} \frac{a + u}{a + 1} \theta^{a+1} - \theta^a + \frac{1 - u}{a + 1} = 0.\end{equation*} This choice of $\theta$ is the best possible. THEOREM 2. If $x_1, x_2, \cdots$ satisfy $| x_n | \leqq 1$ and $E(x_n \mid x_1, \cdots, x_{n - 1}) = 0,$ then for all $N > 0,$ $$\mathrm{Pr}\big\{\big|\frac{x_1 + \cdots + x_n}{n}\big| \geqq \epsilon \text{for some} n \geqq N\big\} \leqq 2_\varphi^N,$$ where $\varphi = (1 + \epsilon)^{-(1+\epsilon)/2}(1 - \epsilon)^{-(1 - \epsilon)/2}.$ This choice of $\varphi$ is, for every $\epsilon$ between 0 and 1, the best possible. Both results are improvements of results of Blackwell [1], and the methods of proof are somewhat similar.

Article information

Source
Ann. Math. Statist., Volume 27, Number 4 (1956), 1170-1174.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728086

Digital Object Identifier
doi:10.1214/aoms/1177728086

Mathematical Reviews number (MathSciNet)
MR83831

Zentralblatt MATH identifier
0073.12503

JSTOR
links.jstor.org

Citation

Weingarten, Harry. On the Probability of Large Deviations for Sums of Bounded Chance Variables. Ann. Math. Statist. 27 (1956), no. 4, 1170--1174. doi:10.1214/aoms/1177728086. https://projecteuclid.org/euclid.aoms/1177728086


Export citation