A New Randomized Pólya Urn Model - Applied Mathematics

AM > Vol.3 No.12, December 2012

A New Randomized Pólya Urn Model ()

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DOI: 10.4236/am.2012.312A292 7,433 Downloads 11,635 Views Citations

Author(s)

Djilali Ait Aoudia, Francois Perron

Affiliation(s)

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

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_nballs and note their colors, say R_nare white and M_n- R_n are black. We return the drawn balls in the urn. Moreover, N_nR_nnew white balls and N_n (M_n- R_n) new black balls are added in the urn. The numbers M_nand N_nare 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

Share and Cite:

D. Aoudia and F. Perron, "A New Randomized Pólya Urn Model," Applied Mathematics, Vol. 3 No. 12A, 2012, pp. 2118-2122. doi: 10.4236/am.2012.312A292.

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