On Optimal Systems

Abstract

For any sequence $x_1, x_2, \cdots$ of chance variables satisfying $| x_n | \leqq 1$ and $E(x_n\| x_1, \cdots, x_{n-1} \leqq -u(\max | x_n\| x_1, \cdots, x_{n-1})$, where $u$ is a fixed constant, $0 < u < 1$, and for any positive number $t$, $\mathrm{Pr} \{ \underset{n}\sup (x_1 + \cdots + x_n) \geqq t\} \leqq \big(\frac{1 - u}{1 + u}\big)^t.$ Equality holds for integral $t$ when $x_1, x_2, \cdots$ are independent with $\mathrm{Pr} \{x_n = 1\} = (1 - u)/2, \quad \mathrm{Pr} \{x_n = -1\} = (1 + u)/2.$ This has a simple interpretation in terms of gambling systems, and yields a new proof of Levy's extension of the strong law of large numbers to dependent variables [2], with an improved estimate for the rate of convergence.