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Refined asymptotics for the composition of cyclic urns. (English) Zbl 1406.60043
Summary: 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.
##### 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)
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##### 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. · Zbl 1296.68039 [2] Chauvin, B., Mailler, C. and Pouyanne, N. (2015) Smoothing equations for large Pólya urns. J. Theor. Probab.28, 923–957. · Zbl 1358.60010 [3] Chern, H.-H., Fuchs, M. and Hwang, H.-K. (2007) Phase changes in random point quadtrees. ACM Trans. Algorithms3, Art. 12, 51 pp. · Zbl 1321.68218 [4] Chern, H.-H. and Hwang, H.-K. (2001) Phase changes in random $$m$$-ary search trees and generalized quicksort. Random Structures Algorithms19, 316–358. · Zbl 0990.68052 [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. · Zbl 1143.68019 [6] Evans, S.N., Grübel, R. and Wakolbinger, A. (2012) Trickle-down processes and their boundaries. Electron. J. Probab.17, 1-58. · Zbl 1246.60100 [7] Freedman, D. A. (1965) Bernard Friedman’s Urn. Ann. Math. Statist.36, no. 3, 956–970. · Zbl 0138.12003 [8] Grübel, R. (2014) Search trees: Metric aspects and strong limit theorems. Ann. Appl. Probab.24, 1269–1297. · Zbl 1294.60009 [9] Janson, S. (1983) Limit theorems for certain branching random walks on compact groups and homogeneous spaces. Ann. Probab.11, 909–930. · Zbl 0544.60022 [10] Janson, S. (2004) Functional limit theorem for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl.110, 177–245. · Zbl 1075.60109 [11] Janson, S. (2006) Congruence properties of depths in some random trees. Alea1, 347–366. · Zbl 1104.60300 [12] Knape, M. and Neininger, R. (2014) Pólya Urns Via the Contraction Method. Combin. Probab. Comput.23, 1148–1186. · Zbl 1301.60012 [13] Mahmoud, H. M. (1992) Evolution of Random Search Trees, John Wiley & Sons, New York. · Zbl 0762.68033 [14] Mailler, C. (2018) Balanced multicolour Pólya urns via smoothing systems analysis. ALEA - Latin American Journal of Probability and Mathematical StatisticsXV, 375–408. · Zbl 1390.60115 [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 Algorithms46, 346–361. · Zbl 1327.68086 [17] Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab.14, 378-418. · Zbl 1041.60024 [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. · Zbl 1096.60007 [19] Pouyanne, N. (2008) An algebraic approach to Pólya processes. Ann. Inst. Henri Poincaré Probab. Stat.44, 293–323.
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