## Electronic Journal of Probability

### Refined asymptotics for the composition of cyclic urns

#### 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

#### 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

#### References

• [1] Bindjeme, P. and Fill, J. A. (2012) Exact $L^2$-Distance from the Limit for QuickSort Key Comparisons (Extended abstract). DMTCS proc. AQ, 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’12), 339–348.
• [2] Chauvin, B., Mailler, C. and Pouyanne, N. (2015) Smoothing equations for large Pólya urns. J. Theor. Probab. 28, 923–957.
• [3] Chern, H.-H., Fuchs, M. and Hwang, H.-K. (2007) Phase changes in random point quadtrees. ACM Trans. Algorithms 3, Art. 12, 51 pp.
• [4] Chern, H.-H. and Hwang, H.-K. (2001) Phase changes in random $m$-ary search trees and generalized quicksort. Random Structures Algorithms 19, 316–358.
• [5] Drmota, M., Janson, S. and Neininger, R. (2008) A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18, 288–333.
• [6] Evans, S.N., Grübel, R. and Wakolbinger, A. (2012) Trickle-down processes and their boundaries. Electron. J. Probab. 17, 1-58.
• [7] Freedman, D. A. (1965) Bernard Friedman’s Urn. Ann. Math. Statist. 36, no. 3, 956–970.
• [8] Grübel, R. (2014) Search trees: Metric aspects and strong limit theorems. Ann. Appl. Probab. 24, 1269–1297.
• [9] Janson, S. (1983) Limit theorems for certain branching random walks on compact groups and homogeneous spaces. Ann. Probab. 11, 909–930.
• [10] Janson, S. (2004) Functional limit theorem for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110, 177–245.
• [11] Janson, S. (2006) Congruence properties of depths in some random trees. Alea 1, 347–366.
• [12] Knape, M. and Neininger, R. (2014) Pólya Urns Via the Contraction Method. Combin. Probab. Comput. 23, 1148–1186.
• [13] Mahmoud, H. M. (1992) Evolution of Random Search Trees, John Wiley & Sons, New York.
• [14] Mailler, C. (2018) Balanced multicolour Pólya urns via smoothing systems analysis. ALEA - Latin American Journal of Probability and Mathematical Statistics XV, 375–408.
• [15] Müller, N. S. and Neininger, R. (2016) The CLT Analogue for Cyclic Urns. Analytic Algorithmics and Combinatorics (ANALCO), 121–127.
• [16] Neininger, R. (2015) Refined Quicksort asymptotics. Random Structures Algorithms 46, 346–361.
• [17] Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14, 378-418.
• [18] Pouyanne, N. (2005) Classification of large Pólya-Eggenberger urns with regard to their asymptotics. 2005 International Conference on Analysis of Algorithms, 275–285 (electronic), Discrete Math. Theor. Comput. Sci. Proc., AD, Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [19] Pouyanne, N. (2008) An algebraic approach to Pólya processes. Ann. Inst. Henri Poincaré Probab. Stat. 44, 293–323.