Electronic Journal of Probability

Refined asymptotics for the composition of cyclic urns

Noela Müller and Ralph Neininger

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 117, 20 pp.

Dates
Received: 30 January 2017
Accepted: 7 November 2018
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1543028707

Digital Object Identifier
doi:10.1214/18-EJP243

Mathematical Reviews number (MathSciNet)
MR3885550

Zentralblatt MATH identifier
07021673

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Pólya urn cyclic urn cyclic group periodicities weak convergence CLT analogue probability metric Zolotarev metric

Rights
Creative Commons Attribution 4.0 International License.

Citation

Müller, Noela; Neininger, Ralph. Refined asymptotics for the composition of cyclic urns. Electron. J. Probab. 23 (2018), paper no. 117, 20 pp. doi:10.1214/18-EJP243. https://projecteuclid.org/euclid.ejp/1543028707


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