Refined Asymptotics for the Composition of Cyclic Urns

TL;DR

This paper investigates the asymptotic behavior of the composition of cyclic urns, revealing normal fluctuations for various sizes and identifying the structure of these fluctuations depending on the number of types.

Contribution

It provides a detailed analysis of the asymptotic fluctuations of cyclic urn compositions, extending known results to larger urn sizes and refining the understanding of fluctuation dimensions.

Findings

Asymptotic normality of fluctuations for 7 ≤ m ≤ 12.

Refined normal fluctuation results for m ≥ 13.

Fluctuation dimension depends on divisibility of m by 6.

Abstract

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.