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

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.