Refined Asymptotics for the Composition of Cyclic Urns

A cyclic urn is an urn model for balls of types $0,\ldots,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then 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 $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector. In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $7\le m\le 12$. For $m\ge 13$ we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations 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.