Journal of Applied Probability

Strong convergence for urn models with reducible replacement policy

R. Abraham, J. S. Dhersin, and B. Ycart

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A multitype urn scheme with random replacements is considered. Each time a ball is picked, another ball is added, and its type is chosen according to the transition probabilities of a reducible Markov chain. The vector of frequencies is shown to converge almost surely to a random element of the set of stationary measures of the Markov chain. Its probability distribution is characterized as the solution to a fixed point problem. It is proved to be Dirichlet in the particular case of a single transient state to which no return is possible. This is no longer the case, however, as soon as returns to transient states are allowed.

Article information

Source
J. Appl. Probab. Volume 44, Number 3 (2007), 652-660.

Dates
First available in Project Euclid: 13 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.jap/1189717535

Digital Object Identifier
doi:10.1239/jap/1189717535

Mathematical Reviews number (MathSciNet)
MR2355582

Zentralblatt MATH identifier
1138.60030

Subjects
Primary: 60F15: Strong theorems

Keywords
Urn model Markov chain strong convergence

Citation

Abraham, R.; Dhersin, J. S.; Ycart, B. Strong convergence for urn models with reducible replacement policy. J. Appl. Probab. 44 (2007), no. 3, 652--660. doi:10.1239/jap/1189717535. https://projecteuclid.org/euclid.jap/1189717535


Export citation

References

  • Bai, Z. D. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Prob. 15, 914--940.
  • Bai, Z. D., Hu, F. and Zhang, L. X. (2002). Gaussian approximation theorems for urn models and their applications. Ann. Appl. Prob. 12, 1149--1173.
  • Benaïm, M., Schreiber, S. and Tarrès, P. (2004). Generalized urn models of evolutionary processes. Ann. Appl. Prob. 14, 1455--1478.
  • Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and product of random matrices. J. Theoret. Prob. 4, 3--36.
  • Delyon, B. (1996). General results on the convergence of stochastic algorithms. IEEE Trans. Automatic Control 41, 1245--1255.
  • Duflo, M. (1997). Random Iterative Models. Springer, New York.
  • Eggenberger, F. and Pólya, G. (1928). Sur l'interprétation de certaines courbes de fréquence. C. R. Acad. Sci. Paris 187, 870--872.
  • Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Prob. 21, 1624--1639.
  • Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn. J. Appl. Prob. 34, 426--435.
  • Higueras, I., Moler, J., Plo, F. and San Miguel, M. (2003). Urn models and differential algebraic equations. J. Appl. Prob. 40, 401--412.
  • Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177--245.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.
  • Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. Springer, New York.
  • Letac, G. (1986). A contraction principle for certain Markov chains and its applications. Contemp. Math. 50, 263--273.
  • Pitman, J. (1996). Some developments of the Blackwell--MacQueen urn scheme. In Statistics, Probability and Game Theory (IMS Lecture Notes Monogr. Ser. 30), Institute of Mathematical Statistics, Hayward, CA, pp. 245\nobreakdash--267.
  • Schreiber, S. (2001). Urn models, replicator processes and random genetic drift. SIAM J. Appl. Math. 61, 2148\nobreakdash--2167.