Electronic Journal of Probability

Refined asymptotics for the composition of cyclic urns

Noela Müller and Ralph Neininger

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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

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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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