• Bernoulli
  • Volume 15, Number 1 (2009), 279-295.

Multicolor urn models with reducible replacement matrices

Arup Bose, Amites Dasgupta, and Krishanu Maulik

Full-text: Open access


Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak limits are known.

Article information

Bernoulli, Volume 15, Number 1 (2009), 279-295.

First available in Project Euclid: 3 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

martingale reducible stochastic replacement matrix urn model variance mixture of normal


Bose, Arup; Dasgupta, Amites; Maulik, Krishanu. Multicolor urn models with reducible replacement matrices. Bernoulli 15 (2009), no. 1, 279--295. doi:10.3150/08-BEJ150.

Export citation


  • [1] Bai, Z.-D. and Hu, F. (2005). Asymptotics in randomized urn models., Ann. Appl. Probab. 15 914–940.
  • [2] Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory., Discrete Math. Theor. Comput. Sci. AG 59–118 (electronic).
  • [3] Freedman, D.A. (1965). Bernard Friedman’s urn., Ann. Math. Statist. 36 956–970.
  • [4] Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn., J. Appl. Probab. 34 426–435.
  • [5] Hall, P. and Heyde, C.C. (1980)., Martingale Limit Theory and Its Application. New York: Academic Press Inc.
  • [6] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns., Stochastic Process. Appl. 110 177–245.
  • [7] Janson, S. (2006). Limit theorems for triangular urn schemes., Probab. Theory Related Fields 134 417–452.
  • [8] Pouyanne, N. (2008). An algebraic approach to Pólya processes., Ann. Inst. H. Poincaré Probab. Statist. 44 293–323.
  • [9] Puyhaubert, V. (2005). Modèles d’urnes et phénomènes de seuil en combinatoire analytique. Ph.D. thesis, École Polytechnique, Palaiseau, France.