A Necessary Condition for Making Money from Fair Games

Abstract

Let $X_1,X_2,\ldots$ be independent random variables such that $X_j$ has distribution $F_{\sigma(j)}$, where $\sigma(j) = 1$ or 2, and the distributions $F_i$ have mean 0. Assume that $F_i$ has a finite $q_i$th moment for some $1 < q_i < 2$. Let $S_n = \sum^n_{j=1}X_j$. We show that if $q_1 + q_2 > 3$, then $\lim\sup P\{S_n > 0\} > 0$ and $\lim\sup P\{S_n < 0\} > 0$ for each sequence $\{\sigma(j)\}$ of ones and twos.