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.
"A martingale approach for Pólya urn processes." Electron. Commun. Probab. 25 1 - 13, 2020. https://doi.org/10.1214/20-ECP321