Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. A classical theorem of Rényi [29] for the number of empty boxes, denoted by μ 0 (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 μ 0 (n, M) tends to infinity with n, then μ 0 (n, M) is asymptotically normally distributed. This result, seldom stated in this form in the literature, was proved by Rényi [29] by dissecting the range of n and M into three different ranges, in each of which a different method of proof was employed. Local limit theorems were later studied by Sevast'yanov and Chistyakov [30] in a rather limited range when both ratios of n/M and M/n remain bounded. Kolchin [22] gave a very detailed study on different approximation theorems. For a fairly complete account of this theory, see Kolchin, Sevast'yanov and Chistyakov [23]. Englund [9] 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 falling into the j th box being p j = p j (M), 0≤j<M p j = 1. Quine and Robinson [27] showed that if p j M is bounded for j = 0, 1, . . . , M − 1 and if the variance of μ 0 (n, M) tends to infinity with n (n is the number of allocations), then the distribution of μ 0 (n, M) is asymptotically normal. They indeed derived a Berry-Esseen bound for the normal approximation of the distribution. Their result remains the strongest of its kind in the literature. Note that the condition p j M = O(1) for all j is one of the essential conditions needed for proving the asymptotic normality of μ 0 (n, M) in many previous papers on multinomial schemes (see Holst [15], Kolchin, Sevast'yanov and Chistyakov [23] ); it implies that the general multinomial scheme studied in the literature is indeed not very far from the equiprobable one. Our major contribution of this paper is to show that this condition can be completely removed. Moreover, under the minimum condition that
For example the work of REF (effectively) provides a local limit theorem for K n provided that the variance tends to infinity.
Local limit theorems for finite and infinite urn models
Ann. Probab
Mathematics