%0 Journal Article
%T A New Randomized Pólya Urn Model
%A Djilali Ait Aoudia
%A Francois Perron
%J Applied Mathematics
%P 2118-2122
%@ 2152-7393
%D 2012
%I Scientific Research Publishing
%R 10.4236/am.2012.312A292
%X <span>In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized P¨®lya urn, that is reinforced every time. A single urn contains </span><i><span>a</span></i><span> white balls, </span><i><span>b</span></i><span> black balls and evolves as follows: at discrete times </span><i><span>n</span></i><span>=1,2,ˇ­,</span><span> we sample</span><i><span> M<sub>n </sub></span></i><span>balls and note their colors, say</span><i><span> R<sub>n </sub></span></i><span>are white and </span><i><span>M<sub>n</sub>- R<sub>n</sub></span></i><span> are black. We return the drawn balls in the urn. Moreover, </span><i><span>N<sub>n</sub>R<sub>n </sub></span></i><span>new white balls and </span><i><span>N<sub>n</sub></span></i><span> (</span><i><span>M<sub>n</sub>- R<sub>n</sub></span></i><span>)</span><i><span> </span></i><span>new black balls are added in the urn. The numbers</span><i><span> M<sub>n </sub></span></i><span>and</span><i><span> N<sub>n </sub></span></i><span>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 </span><span>{0,1} </span><span>are given.</span><span></span>
%K Urn Model
%K Martingale
%K Asymptotic Exchangeability
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=26054