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 Mn balls and note their colors, say Rn are white and Mn- Rn are black. We return the drawn balls in the urn. Moreover, NnRn new white balls and Nn (Mn- Rn)new black balls are added in the urn. The numbers Mn and Nn 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.
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