A New Two-Urn Model of P?lya-Type

Abstract

In this talk, we propose a new two-urn model of P?lya type as follows. There are two urns, urn A and urn B. Suppose, at the beginning, urn A contains rA red and wA white balls and urn B contains rB red and wB white balls. We first draw m balls from urn A and note their colors, say i red and m-i white balls. Then return them into urn A and add bi red and b(m-i) white balls into urn B. Next, we draw ? balls from urn B and note their colors, say j red and ?-j white balls. Then return them into urn B and add aj red and a(?- j) white balls into urn A. Repeat the above action n times and let Xn be the ratio of red balls in urn A and Yn the ratio of red balls in urn B. We use martingale and stochastic approximation to show that Xn and Yn converge almost surely to the same limit. Moreover, in some cases, we can derive the closed form of the limit of the expectation of Xn.Keyword: urn model; martingale; stochastic approximation.