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Generalized Friedman urn is one of the simplest and most useful modelsconsidered in probability theory. Since Athreya and Ney (1972) showed thealmost sure convergence of urn proportions in a randomized urn model withirreducible replacement matrix under the $L\log L$ moment assumption, thisassumption has been regarded as the weakest moment assumption, but thenecessary has never been shown. In this paper, we study the strong and weakconvergence of generalized Friedman urns. It is proved that, when the randomreplacement matrix is irreducible in probability, the sufficient and necessarymoment assumption for the almost sure convergence of the urn proportions isthat the expectation of the replacement matrix is finite, which is lessstringent than the $L\log L$ moment assumption, and when the replacement isreducible, the $L\log L$ moment assumption is the weakest sufficient condition.The rate of convergence and the strong and weak convergence of non-homogenousgeneralized Friedman urns are also derived.

Convergence of randomized urn models with irreducible and reducible replacement policy