A New Randomized Pólya Urn Model

Djilali Ait Aoudia, Francois Perron

Université de Montréal, Département de mathématiques et de statistique, Montréal, Canada.

DOI: 10.4236/am.2012.312A292                

Abstract

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains a white balls, b black balls and evolves as follows: at discrete times n=1,2,…, we sample M_n balls and note their colors, say R_n are white and M_n- R_n are black. We return the drawn balls in the urn. Moreover, N_nR_n new white balls and N_n (M_n- R_n) new black balls are added in the urn. The numbers M_n and N_n are random variables. We show that the proportions of white balls forms a bounded martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set {0,1} are given.

Keywords

Urn Model; Martingale; Asymptotic Exchangeability

Conflicts of Interest

The authors declare no conflicts of interest.

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