Theoretical analysis demonstrates asymptotic behavior and new almost sure properties in Pólya urn processes, highlighting the effectiveness of direct martingale methods.
This paper is devoted to a direct martingale approach for Pólya urn models asymptotic behaviour. A Pólya process is said to be small when the ratio of its replacement 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 urn processes. We also provide new almost sure properties for small urn processes.
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Lucile Laulin (2020) studied this question.
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