Electronic Journal of Probability

Measure-valued Pólya urn processes

Cécile Mailler and Jean-François Marckert

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A Pólya urn process is a Markov chain that models the evolution of an urn containing some coloured balls, the set of possible colours being $\{1,\ldots ,d\}$ for $d\in \mathbb{N} $. At each time step, a random ball is chosen uniformly in the urn. It is replaced in the urn and, if its colour is $c$, $R_{c,j}$ balls of colour $j$ are also added (for all $1\leq j\leq d$).

We introduce a model of measure-valued processes that generalises this construction. This generalisation includes the case when the space of colours is a (possibly infinite) Polish space $\mathcal P$. We see the urn composition at any time step $n$ as a measure ${\cal M}_n$ – possibly non atomic – on $\mathcal P$. In this generalisation, we choose a random colour $c$ according to the probability distribution proportional to ${\cal M}_n$, and add a measure ${\cal R}_c$ in the urn, where the quantity ${\cal R}_c(B)$ of a Borel set $B$ models the added weight of “balls” with colour in $B$.

We study the asymptotic behaviour of these measure-valued Pólya urn processes, and give some conditions on the replacements measures $({\cal R}_c,c\in \mathcal P)$ for the sequence of measures $({\cal M}_n, n\geq 0)$ to converge in distribution, possibly after rescaling. For certain models, related to branching random walks, $({\cal M}_n, n\geq 0)$ is shown to converge almost surely under some moment hypothesis; a particular case of this last result gives the almost sure convergence of the (renormalised) profile of the random recursive tree to a standard Gaussian.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 26, 33 pp.

Received: 11 October 2016
Accepted: 8 March 2017
First available in Project Euclid: 21 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Pólya urns limit theorems branching Markov chains branching random walks

Creative Commons Attribution 4.0 International License.


Mailler, Cécile; Marckert, Jean-François. Measure-valued Pólya urn processes. Electron. J. Probab. 22 (2017), paper no. 26, 33 pp. doi:10.1214/17-EJP47. https://projecteuclid.org/euclid.ejp/1490061796

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  • [1] M. Albenque and J.-F. Marckert. Some families of increasing planar maps. Electronic Journal of Probability, 13:1624–1671, 2008.
  • [2] K. B. Athreya and P. E. Ney. Branching Processes. Springer-Verlag/Berlin, 1972.
  • [3] A. Bandyopadhyay and D. Thacker. A new approach to Pólya urn schemes and its infinite color generalization. (arXiv:1606.05317)
  • [4] A. Bandyopadhyay and D. Thacker. On Pólya urn schemes with infinitely many colors. Bernoulli journal. To appear (arXiv:1303.7374).
  • [5] A. Bandyopadhyay and D. Thacker. Rate of convergence and large deviation for the infinite color Pólya urn schemes. Statistics & Probability Letters, 92:232–240, 2014.
  • [6] F. Bergeron, P. Flajolet, and B. Salvy. Varieties of increasing trees, 1992. Available at https://hal.inria.fr/inria-00074977/document.
  • [7] P. Berti, L. Pratelli, and P. Rigo. Almost sure weak convergence of random probability measures. Stochastics, 78:91–97, 2006.
  • [8] J. D. Biggins. Uniform convergence of martingales in the branching random walk. Annals of Probability, 20(1):137–151, 1992.
  • [9] N. Broutin and L. Devroye. Large deviations for the weighted height of an extended class of trees. Algorithmica, 46:271–297, 2006.
  • [10] S. D. Chatterji. Martingales of Banach-valued random variables. Bulletin of the American Mathematical Society, 66(5):395–398, 09 1960.
  • [11] B. Chauvin, M. Drmota, and J. Jabbour-Hattab. The profile of binary search trees. The Annals of Applied Probability, 11(4):1042–1062, 11 2001.
  • [12] B. Chauvin, T. Klein, J.-F. Marckert, and A. Rouault. Martingales and profile of binary search trees. Electronic Journal of Probability, 10:420–435, 2005.
  • [13] B. Chauvin, C. Mailler, and N. Pouyanne. Smoothing equations for large Pólya urns. Journal of Theoretical Probability, 28:923–957, 2015.
  • [14] L. Devroye. Applications of the theory of records in the study of random trees. Acta Informatica, 26(1):123–130, 1988.
  • [15] L. Devroye and B. Reed. On the variance of the height of random binary search trees. SIAM Journal on Computing, pages 1157–1162, 1995.
  • [16] R. P. Dobrow. On the distribution of distances in recursive trees. Journal of Applied Probability, 33:749–757, 1996.
  • [17] M. Drmota and B. Gittenberger. On the profile of random trees. Random Structures and Algorithms, 10:421–451, 1997.
  • [18] M. Drmota and H. Hsien-Kuei. Profiles of random trees: correlation and width of random recursive trees and binary search trees. Advances in Applied Probability, 37:321–341, 2005.
  • [19] M. Drmota, S. Janson, and R. Neininger. A functional limit theorem for the profile of search trees. Annals of Applied Probability, 18:288,333, 2008.
  • [20] E. Fekete. Branching random walks on binary search trees: convergence of the occupation measure. ESAIM: Probability and Statistics, 14:286–298, Oct. 2010.
  • [21] P. Flajolet, J. Gabarró, and H. Pekari. Analytic Urns. Annals of Probability, 2005.
  • [22] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge university press, 2009.
  • [23] M. Fuchs, H.-K. Hwang, and R. Neininger. Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica, 46(3):367–407, 2006.
  • [24] S. Janson. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Processes and Applications, 110(2):177–245, 2004.
  • [25] A. Joffe, L. Le Cam, and J. Neveu. Sur la loi des grands nombres pour des variables aléatoires de Bernoulli attachées à un arbre dyadique. Comptes Rendus de l’Académie des Sciences de Paris, Série A277, pages 963–964, 1973.
  • [26] N. L. Johnson and S. Kotz. Urn models and their applications. Wiley and sons, 1997.
  • [27] Z. Katona. Width of a scale-free tree. Journal of Applied Probability, 42:839–850, 2005.
  • [28] M. Knape and R. Neininger. Pólya urns via the contraction method. Combinatorics Probability and Computing, 23(6):1148–1186, 2014.
  • [29] M. Kuba and S. G. Wagner. On the distribution of depths in increasing trees. Electronic Journal of Combinatorics, 17(1), 2010.
  • [30] H. Mahmoud and B. Pittel. On the most probable shape of a search tree grown from a random permutation. SIAM Journal on Algebraic Discrete Methods, 5(1):69–81, 1984.
  • [31] J.-F. Marckert. The rotation correspondence is asymptotically a dilatation. Random Structures and Algorithms, 24(2):118–132, 2004.
  • [32] G. Pisier. Martingales in Banach spaces. Cambridge University Press, 2016. The authors refer to the mini-course version available at https://webusers.imj-prg.fr/~gilles.pisier/ihp-pisier.pdf.
  • [33] E.-M. Schopp. A functional limit theorem for the profile of $b$-ary trees. The Annals of Applied Probability, 20(3):907–950, 2010.
  • [34] H. Sulzbach. A functional limit law for the profile of plane-oriented recursive trees. DMTCS Proceedings, 0(1), 2008.