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Central Limit Theorems for Generalized Polya urn Models

I. Higueras, J. Moler, F. Plo and M. San Miguel
Journal of Applied Probability
Vol. 43, No. 4 (Dec., 2006), pp. 938-951
Stable URL: http://www.jstor.org/stable/27595791
Page Count: 14
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Central Limit Theorems for Generalized Polya urn Models
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Abstract

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

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