Bernard Friedman's Urn

Abstract

The case $\beta = 0$ is the famous Polya (1931) Urn; a discussion of its elementary properties can be found in (Feller, 1960, Chapter IV) and (Frechet, 1943). These facts about the Polya Urn are a classical part of the oral tradition, although some have yet to appear in print (see Blackwell and Kendall, 1964). The fractions $(W_n + B_n)^{-1}W_n$ converge with probability 1 to a limiting random variable $Z$, which has a beta distribution with parameters $W_0/\alpha, B_0/\alpha$. Given $Z$, the successive differences $W_{n + 1} - W_n :n \geqq 0$ are conditionally independent and identically distributed, being $\alpha$ with probability $Z$ and 0 with probability $1 - Z$. Proofs are in Section 2. If $\beta > 0$, the situation is radically different. No matter how large $\alpha$ is in comparison with $\beta$, the fractions $(W_n + B_n)^{-1}W_n$ converge to $\frac{1}{2}$ with probability 1. This seemingly paradoxical result can be sharpened in several ways. Abbreviate $\rho$ for $(\alpha + \beta)^{-1}(\alpha - \beta)$. If $\rho > \frac{1}{2}$, it is proved in Section 3 that $(W_n + B_n)^{-\rho}. (W_n - B_n)$ converges with probability 1 to a nondegenerate limiting random variable. This result in turn fails for $\rho \leqq \frac{1}{2}$. If $0 0$. Suppose first $\alpha > \beta$. If $0 \leqq x \leqq 1$ and $P\lbrack\lim \sup (W_n + B_n)^{-1}W_n \leqq x\rbrack = 1$, by an easy variation of the Strong Law, with probability 1, in $N$ trials there will be at most $Nx + o(N)$ drawings of a white ball; so at least $N(1 - x) - o(N)$ drawings of black. Therefore, with probability 1, $\lim \sup (W_n + B_n)^{-1}B_n$ is bounded above by $\lim_{N\rightarrow\infty}\{\alpha\lbrack Nx + o(N)\rbrack + \beta\lbrack N(1 - x) - o(N)\rbrack\}/N(\alpha + \beta)$ or $(\alpha + \beta)^{-1}\lbrack\beta + (\alpha - \beta)x\rbrack$. Starting with $x = 1$ and iterating, $P\lbrack\lim \sup (W_n + B_n)^{-1} \leqq \frac{1}{2}\rbrack = 1$ follows. Interchange white and black to complete the proof for $\alpha > \beta$. If $\alpha < \beta$, and $P\lbrack\lim \sup (W_n + B_n)^{-1}W_n \leqq x\rbrack = 1$, then a similar argument shows $P\lbrack\lim \sup (W_n + B_n)^{-1}B_n \leqq (\alpha + \beta)^{-1}. (\alpha + (\beta - \alpha)x)\rbrack = 1$. The argument proceeds as before, except both colors must be considered simultaneously.