  Generalized Friedman urn is one of the simplest and most useful models
considered in probability theory. Since Athreya and Ney (1972) showed the
almost sure convergence of urn proportions in a randomized urn model with
irreducible replacement matrix under the $L\log L$ moment assumption, this
assumption has been regarded as the weakest moment assumption, but the
necessary has never been shown. In this paper, we study the strong and weak
convergence of generalized Friedman urns. It is proved that, when the random
replacement matrix is irreducible in probability, the sufficient and necessary
moment assumption for the almost sure convergence of the urn proportions is
that the expectation of the replacement matrix is finite, which is less
stringent than the $L\log L$ moment assumption, and when the replacement is
reducible, the $L\log L$ moment assumption is the weakest sufficient condition.
The rate of convergence and the strong and weak convergence of non-homogenous
generalized Friedman urns are also derived.
