The CLT Analogue for Cyclic Urns

A cyclic urn is an urn model for balls of types 0, . . . ,m − 1 where in each draw the ball drawn, say of type j, is returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all m ≥ 7. However, they are of maximal dimension m− 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace. MSC2010: 60F05, 60F15, 60C05, 60J10.