Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reinforced random walks and random distributions

Authors: R. Daniel Mauldin and S. C. Williams
Journal: Proc. Amer. Math. Soc. 110 (1990), 251-258
MSC: Primary 60A99; Secondary 60C05, 60G09, 62A15
MathSciNet review: 1012934
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a classical Polya urn process on a complete binary tree. This process generates an exchangeable sequence of random variables $ {Z_n}$, with values in $ [0,1]$. It is shown that the empirical distribution $ ^\char93 \{ i \leq n:{Z_i} \leq s\} /n$ converges weakly and the distribution of this limit is the same as a standard Dubins-Freedman random distribution. As an application, the variance of the first moment of these Dubins-Freedman distributions is calculated.

References [Enhancements On Off] (What's this?)

  • [1] D. J. Aldous (1985), Exchangeability and related topics, Ecole d'ete de Probabilites de Saint-Flour XIII-1983, Lecture Notes in Math., vol. 1117, Springer-Verlag, New York, 1985, pp. 1-198. MR 883646 (88d:60107)
  • [2] D. Blackwell and D. Kendall, The Martin boundary for Polya's urn scheme, and an application to stochastic population growth, J. Appl. Probab. 1 (1964), 284-296. MR 0176518 (31:790)
  • [3] P. Diaconis and D. A. Freedman, de Finetti's theorem for Markov chains, Ann. Probab. 8 (1980), 115-130. MR 556418 (81f:60090)
  • [4] L. E. Dubins and D. A. Freedman, Random distribution functions, Proceedings, Fifth Berkeley Symp. on Math. Statistics and Probability (L. M. LeCam and J. Neyman, eds.), Univ. of California Press, Berkeley/Los Angeles, 1967, pp. 183-214. MR 0214109 (35:4960)
  • [5] D. A. Freedman (1965), Bernard Friedman's urn, Ann. Math. Statist. 36, 956-970. MR 0177432 (31:1695)
  • [6] S. Graf, R. D. Mauldin and S. C. Williams, Random homeomorphisms, Adv. in Math. 60 (1986), 239-359. MR 848153 (87k:60010)
  • [7] E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470-501. MR 0076206 (17:863g)
  • [8] B. M. Hill, D. Lane and W. Sudderth, Exchangeable urn processes, Ann. Probab. 15 (1987), 1586-1592. MR 905350 (88g:60094)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60A99, 60C05, 60G09, 62A15

Retrieve articles in all journals with MSC: 60A99, 60C05, 60G09, 62A15

Additional Information

Keywords: Polya urn process, exchangable, empirical distribution, random distribution
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society