Electronic Communications in Probability

Random replacements in Pólya urns with infinitely many colours

Svante Janson

Full-text: Open access


We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times [0,1]$.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 23, 11 pp.

Received: 27 November 2017
Accepted: 3 April 2019
First available in Project Euclid: 30 April 2019

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability

Pólya urn infinite Pólya urn random replacements

Creative Commons Attribution 4.0 International License.


Janson, Svante. Random replacements in Pólya urns with infinitely many colours. Electron. Commun. Probab. 24 (2019), paper no. 23, 11 pp. doi:10.1214/19-ECP226. https://projecteuclid.org/euclid.ecp/1556589626

Export citation


  • [1] Antar Bandyopadhyay and Debleena Thacker, Rate of convergence and large deviation for the infinite color Pólya urn schemes. Statist. Probab. Lett. 92 (2014), 232–240.
  • [2] Antar Bandyopadhyay and Debleena Thacker, Pólya urn schemes with infinitely many colors. Bernoulli 23 (2017), no. 4B, 3243–3267.
  • [3] Antar Bandyopadhyay and Debleena Thacker, A new approach to Pólya urn schemes and its infinite color generalization. Preprint, 2016. arXiv:1606.05317
  • [4] David Blackwell and James B. MacQueen, Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 (1973), 353–355.
  • [5] Donald L. Cohn, Measure Theory, Birkhäuser, Boston, 1980.
  • [6] F. Eggenberger and G. Pólya, Über die Statistik verketteter Vorgänge. Zeitschrift Angew. Math. Mech. 3 (1923), 279–289.
  • [7] Svante Janson, Gaussian Hilbert Spaces. Cambridge Univ. Press, Cambridge, UK, 1997.
  • [8] Svante Janson, Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110 (2004), 177–245.
  • [9] Svante Janson and Lutz Warnke. On the critical probability in percolation. Electronic J. Probability, 23 (2018), paper no. 1, 25 pp.
  • [10] Olav Kallenberg. Foundations of Modern Probability. 2nd ed., Springer, New York, 2002.
  • [11] Olav Kallenberg. Random Measures, Theory and Applications. Springer, Cham, Switzerland, 2017.
  • [12] Hosam M Mahmoud, Pólya urn models. CRC Press, Boca Raton, FL, 2009.
  • [13] Cécile Mailler and Jean-François Marckert. Measure-valued Pólya urn processes. Electron. J. Probab. 22 (2017), no. 26, 1–33.
  • [14] A. A. Markov, Sur quelques formules limites du calcul des probabilités. (Russian.) Bulletin de l’Académie Impériale des Sciences 11 (1917), no. 3, 177–186.
  • [15] K. R. Parthasarathy, Probability Measures on Metric Spaces. Academic Press, New York, 1967.
  • [16] Jim Pitman. Combinatorial Stochastic Processes. École d’Été de Probabilités de Saint-Flour XXXII – 2002. Lecture Notes in Math. 1875, Springer, Berlin, 2006.
  • [17] G. Pólya, Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1 (1930), no. 2, 117–161.