Generalized Friedman urn is one of the simple and 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 by applying the $L\log L$ criteria for the convergence of a branching process, this assumption has been regarded as the weakest moment assumption. But the necessary of the the $L\log L$ moment assumption has never been shown though a lot of studies can be found in literatures for various urn models. In this talk, we will present the strong convergence of generalized randomized 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 a weakest sufficient condition. The rate of convergence and the central limit theorem are also discussed.