Bernard Friedman's Urn

The case = 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 converge with probability 1 to a limiting random variable Z, which has a beta distribution with parameters W₀/, B₀/. Given Z, the successive differences W₍ + ₁ - Wₙ: n 0 are conditionally independent and identically distributed, being with probability Z and 0 with probability 1 - Z. Proofs are in Section 2. If > 0, the situation is radically different. No matter how large is in comparison with, the fractions (Wₙ + Bₙ) ^-1Wₙ converge to 12 with probability 1. This seemingly paradoxical result can be sharpened in several ways. Abbreviate for (+) ^-1 (-). If > 12, it is proved in Section 3 that (Wₙ + Bₙ) ^-. (Wₙ - Bₙ) converges with probability 1 to a nondegenerate limiting random variable. This result in turn fails for 12. If 0 0. Suppose first >. If 0 x 1 and P (Wₙ + Bₙ) ^-1Wₙ x = 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, (Wₙ + Bₙ) ^-1Bₙ is bounded above by ₍\ Nx + o (N) + N (1 - x) - o (N) \/N (+) or (+) ^-1 + (-) x. Starting with x = 1 and iterating, P (Wₙ + Bₙ) ^-1 12 = 1 follows. Interchange white and black to complete the proof for >. If <, and P (Wₙ + Bₙ) ^-1Wₙ x = 1, then a similar argument shows P (Wₙ + Bₙ) ^-1Bₙ (+) ^-1. (+ (-) x) = 1. The argument proceeds as before, except both colors must be considered simultaneously.