The CLT Analogue for Cyclic Urns

TL;DR

This paper investigates the asymptotic behavior of cyclic urn models, revealing that fluctuations are normally distributed for all sizes, with their dimensionality depending on divisibility properties of the number of types.

Contribution

It extends the CLT analogue to cyclic urns for all sizes, detailing the fluctuation structure and its dependence on divisibility conditions.

Findings

Fluctuations are asymptotically normal for all m ≥ 7.

Maximal fluctuation dimension is m-1 unless m is divisible by 6.

For multiples of 6, fluctuations are confined to a 2D subspace.

Abstract

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.