Abstract
The asymptotic distributions of the size distributions of the symmetric Dirichlet-Multinomial urn model are investigated. By allowing the number of ballsn and number of urnsm to go to infinity at different rates we can get both Poisson limit distributions and normal limit distributions. In either case some local limit theorems are obtained. We also consider a more general urn scheme where the number of balls is considered as a random variable depending on the number of urns; some interesting limit theorems are also obtained in this context.
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