Abstract
This paper is devoted to a direct martingale approach for Polya urn models asymptotic behaviour. A Polya process is said to be small when the ratio of its remplacement matrix eigenvalues is less than or equal to 1/2, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urns processes. We also provide new almost sure properties for small urns processes.
Highlights
At the inital time n = 0, an urn is filled with α ≥ 0 red balls and β ≥ 0 white balls.at any time n ≥ 1 one ball is drawn randomly from the urn and its color observed.If it is red it is returned to the urn together with a additional red balls and b ≥ 0 white ones
This paper is devoted to a direct martingale approach for Pólya urn models asymptotic behaviour
We find again some well-known results on the asymptotic behaviour for small and large urn processes
Summary
At any time n ≥ 1 one ball is drawn randomly from the urn and its color observed. If it is red it is returned to the urn together with a additional red balls and b ≥ 0 white ones. We establish the almost sure convergence and the asymptotic normality for this martingale.
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