The Annals of Applied Probability

Limit distributions for large Pólya urns

Brigitte Chauvin, Nicolas Pouyanne, and Reda Sahnoun

Full-text: Open access


We consider a two-color Pólya urn in the case when a fixed number S of balls is added at each step. Assume it is a large urn that is, the second eigenvalue m of the replacement matrix satisfies 1∕2<mS≤1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order nmS. The object of interest is the limit distribution of this random term.

The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree m. The limit laws appear to constitute a new family of probability densities supported by the whole real line.

Article information

Ann. Appl. Probab., Volume 21, Number 1 (2011), 1-32.

First available in Project Euclid: 17 December 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05D40: Probabilistic methods

Pólya urn urn model martingale characteristic function embedding in continuous time multitype branching process Abelian integrals over Fermat curves


Chauvin, Brigitte; Pouyanne, Nicolas; Sahnoun, Reda. Limit distributions for large Pólya urns. Ann. Appl. Probab. 21 (2011), no. 1, 1--32. doi:10.1214/10-AAP696.

Export citation


  • [1] Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 1801–1817.
  • [2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [3] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [4] Chaumont, L. and Yor, M. (2003). Exercises in Probability. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge Univ. Press, Cambridge.
  • [5] Chauvin, B. and Pouyanne, N. (2004). m-ary search trees when m≥27: A strong asymptotics for the space requirements. Random Structures Algorithms 24 133–154.
  • [6] Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Probab. 33 1200–1233.
  • [7] Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities. Discrete Math. Theor. Comput. Sci. Proc., AG 59–118. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [8] Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn. J. Appl. Probab. 34 426–435.
  • [9] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.
  • [10] Janson, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417–452.
  • [11] Lukacs, E. (1960). Characteristic Functions. Griffin’s Statistical Monographs and Courses 5. Hafner Publishing, New York.
  • [12] Mahmoud, H. M. (2009). Pólya Urn Models. Texts in Statistical Science Series. CRC Press, Boca Raton, FL.
  • [13] Pólya, G. (1931). Sur quelques points de la théorie des probabilités. Ann. Inst. Henri Poincaré 1 117–161.
  • [14] Pouyanne, N. (2008). An algebraic approach to Pólya processes. Ann. Inst. Henri Poincaré Probab. Statist. 44 293–323.
  • [15] Sahnoun, R. (2010). On the distributions of triangular Pólya urn processes. Preprint.
  • [16] Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Monographs and Textbooks in Pure and Applied Mathematics 259. Marcel Dekker, New York.
  • [17] Saks, S. and Zygmund, A. (1971). Analytic Functions, 3rd ed. In Monografie Matematyczne, Tom XXVIII, Warszawa, 1952. Elsevier, New York.