Refined asymptotics for the composition of cyclic urns.

*(English)*Zbl 1406.60043Summary: A cyclic urn is an urn model for balls of types \(0,\dots ,m-1\). 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 \(2\leq m\leq 6\). For \(m\geq 7\) 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 \(7\leq m\leq 12\). For \(m\geq 13\) 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.

In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all \(7\leq m\leq 12\). For \(m\geq 13\) 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.

##### MSC:

60F05 | Central limit and other weak theorems |

60F15 | Strong limit 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
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\textit{N. Müller} and \textit{R. Neininger}, Electron. J. Probab. 23, Paper No. 117, 20 p. (2018; Zbl 1406.60043)

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