Bernard Friedman's Urn

Abstract

The case β=0\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 (Wn+Bn)−1Wn(W_n + B_n)^{-1}W_n converge with probability 1 to a limiting random variable ZZ, which has a beta distribution with parameters W0/α,B0/αW_0/\alpha, B_0/\alpha. Given ZZ, the successive differences Wn+1−Wn:n≧0W_{n + 1} - W_n :n \geqq 0 are conditionally independent and identically distributed, being α\alpha with probability ZZ and 0 with probability 1−Z1 - Z. Proofs are in Section 2. If β>0\beta > 0, the situation is radically different. No matter how large α\alpha is in comparison with β\beta, the fractions (Wn+Bn)−1Wn(W_n + B_n)^{-1}W_n converge to 12\frac{1}{2} with probability 1. This seemingly paradoxical result can be sharpened in several ways. Abbreviate ρ\rho for (α+β)−1(α−β)(\alpha + \beta)^{-1}(\alpha - \beta). If ρ>12\rho > \frac{1}{2}, it is proved in Section 3 that (Wn+Bn)−ρ.(Wn−Bn)(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 ρ≦12\rho \leqq \frac{1}{2}. If 0 < \rho \leqq \frac{1}{2}, the sequence (Wn+Bn)−ρ(Wn−Bn)(W_n + B_n)^{-\rho}(W_n - B_n) has plus infinity for superior limit and minus infinity for inferior limit, with probability 1. If ρ<0\rho < 0, the sequence (Wn−Bn)(W_n - B_n) has plus infinity for superior limit and minus infinity for inferior limit, with probability 1. In both cases, the tail σ\sigma-field of (Wn,Bn):n≧0(W_n, B_n) :n \geqq 0 is trivial. If ρ<12\rho <0\rho < 0. Since (Wn+Bn)−1Wn(W_n + B_n)^{-1}W_n converges to 12\frac{1}{2}, therefore Wn−BnW_n - B_n is asymptotically like the sum of nn independent random variables, each equal to +1+1 with probability 12\frac{1}{2} and −1-1 with probability 12\frac{1}{2}. It is tempting to conclude that the distribution of n−12(Wn−Bn)n^{-\frac{1}{2}}(W_n - B_n) converges to normal with mean 0 and variance 1. From the preceding paragraph, however, the asymptotic variance is 13\frac{1}{3}. There is an even more startling difference between the asymptotic behavior of (Wn−Bn):n≧0(W_n - B_n) : n \geqq 0 and that of a coin-tossing game. Let Xn:n≧1X_n :n \geqq 1 be independent and ±1\pm 1 with probability 12\frac{1}{2} each. Let Sn=X1+⋯+Xn,Sj/n,n=n−12SjS_n = X_1 + \cdots + X_n, S_{j/n,n} = n^{-\frac{1}{2}}S_j. Define St,nS_{t,n} for 0≦t≦10 \leqq t \leqq 1 and ntnt not integral by linear interpolation. By the celebrated Invariance Principle of Donsker (1951), the law of {St,n:0≦t≦1}\{S_{t,n} :0 \leqq t \leqq 1\} converges in a strong way to the law of a Brownian motion. However, for ρ<12\rho 0\beta > 0. Suppose first α>β\alpha > \beta. If 0≦x≦10 \leqq x \leqq 1 and P[limsup(Wn+Bn)−1Wn≦x]=1P\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 NN trials there will be at most Nx+o(N)Nx + o(N) drawings of a white ball; so at least N(1−x)−o(N)N(1 - x) - o(N) drawings of black. Therefore, with probability 1, limsup(Wn+Bn)−1Bn\lim \sup (W_n + B_n)^{-1}B_n is bounded above by limN→∞{α[Nx+o(N)]+β[N(1−x)−o(N)]}/N(α+β)\lim_{N\rightarrow\infty}\{\alpha\lbrack Nx + o(N)\rbrack + \beta\lbrack N(1 - x) - o(N)\rbrack\}/N(\alpha + \beta) or (α+β)−1[β+(α−β)x](\alpha + \beta)^{-1}\lbrack\beta + (\alpha - \beta)x\rbrack. Starting with x=1x = 1 and iterating, P[limsup(Wn+Bn)−1≦12]=1P\lbrack\lim \sup (W_n + B_n)^{-1} \leqq \frac{1}{2}\rbrack = 1 follows. Interchange white and black to complete the proof for α>β\alpha > \beta. If α < \beta, and P[limsup(Wn+Bn)−1Wn≦x]=1P\lbrack\lim \sup (W_n + B_n)^{-1}W_n \leqq x\rbrack = 1, then a similar argument shows P[limsup(Wn+Bn)−1Bn≦(α+β)−1.(α+(β−α)x)]=1P\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.