A martingale approach for Polya urn processes

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 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.