The CLT Analogue for Cyclic Urns

A cyclic urn is an urn model for balls of types $0,\ldots,m-1$ 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 $2\le m\le 6$. For $m\ge 7$ 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 $m\ge 7$. 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.Comment: Extended abstract to be replaced later by a full versio

Refined asymptotics for the number of leaves of random point quadtrees

In the early 2000s, several phase change results from distributional convergence to distributional non-convergence have been obtained for shape parameters of random discrete structures. Recently, for those random structures which admit a natural martingale process, these results have been considerably improved by obtaining refined asymptotics for the limit behavior. In this work, we propose a new approach which is also applicable to random discrete structures which do not admit a natural martingale process. As an example, we obtain refined asymptotics for the number of leaves in random point quadtrees. More applications, for example to shape parameters in generalized m-ary search trees and random gridtrees, will be discussed in the journal version of this extended abstract