1 8 A pr 2 00 6 Local limit theorems for finite and infinite urn models

A classical theorem of Rényi [26] for the number of empty boxes, denoted by μ0(n,M), in a sequence of n random allocations of indistinguishable balls into M boxes with equal probability 1/M , can be stated as follows: If the variance of μ0(n,M) tends to infinity with n then μ0(n,M) is asymptotically normally distributed. This result, seldom stated in this form in the literature, was proved by Rényi [26] by dissecting the range of n and M into three different ranges and in each of which a different method of proof was employed. Local limit theorems were later studied by Sevastyanov and Chistyakov [27] in a rather limited range when both ratios of n/M and M/n remain bounded. Kolchin [18] gave a rather detailed study on different approximation theorems. For a rather complete account of this theory, see Kolchin et al. [19]. Englund [8] later derived an explicit Berry– Esseen bound. Multinomial extension of the problem was studied by many authors. In this scheme, balls are successively thrown into M boxes, the probability of each ball