Abstract

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Polya 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.

Highlights

  • We propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time

  • Nn Rn new white balls and Nn M n Rn new black balls are added in the urn

  • We show that the proportions of white balls forms a bounded martingale sequence which converges almost surely

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Summary

Introduction

We propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn that is reinforced every time. This is a generalization of the urn model considered in [10]. A single urn contains a 0 white balls and b 0 black. Necessary and sufficient conditions for the limit to concentrate on the set 0,1 are given. We conclude the paper by proving that interacting reinforced urn process are asymptotically exchangeable

Model Description and Notation
Martingale Property
Zn n n2 1M
F 2 n 1 G a a b
Asymptotic Exchangeability
Conclusions
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