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Probability Theory and Related Fields

, Volume 134, Issue 3, pp 417–452 | Cite as

Limit theorems for triangular urn schemes

  • Svante JansonEmail author
Article

Abstract.

We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depending on the ratio of the diagonal elements in the replacement matrix; the limit laws include normal, stable and Mittag-Leffler distributions as well as some less familiar ones. The results are in some cases similar to, but in other cases strikingly different from, the results for irreducible replacement matrices.

Keywords

Black Ball Characteristic Function Limit Theorem Asymptotic Normality Probability Generate Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York, 1972Google Scholar
  2. 2.
    Athreya, K.B., Karlin, S.: Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39, 1801–1817 (1968)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Athreya, K.B., Ney, P.E.: Branching Processes. Springer-Verlag, Berlin, 1972Google Scholar
  4. 4.
    Bagchi, A., Pal, A.K.: Asymptotic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Methods 6(3), 394–405 (1985)MathSciNetGoogle Scholar
  5. 5.
    Bernstein, S.: Nouvelles applications des grandeurs aléatoires presqu'indépendantes. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 4, 137–150 (1940)zbMATHGoogle Scholar
  6. 6.
    Drmota, M., Vatutin, V.: Limiting distributions in branching processes with two types of particles. In: Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., 84, Springer, New York, 1997, pp. 89–110Google Scholar
  7. 7.
    Eggenberger, F., Pólya, G.: Über die Statistik verketteter Vorgänge. Zeitschrift Angew. Math. Mech. 3, 279–289 (1923)CrossRefGoogle Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. I. Second edition, Wiley, New York, 1957Google Scholar
  9. 9.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. Second edition, Wiley, New York, 1971Google Scholar
  10. 10.
    Flajolet, P., Gabarró, J., Pekari, H.: Analytic urns. Preprint, 2003. Available from http://algo.inria.fr/flajolet/Publications/publist.html (The original version; not the revised!)
  11. 11.
    Flajolet, P., Puyhaubert, V.: In preparationGoogle Scholar
  12. 12.
    Friedman, B.: A simple urn model. Commun. Pure Appl. Math. 2, 59–70 (1949)zbMATHCrossRefGoogle Scholar
  13. 13.
    Jagers, P.: Branching Processes with Biological Applications. Wiley, Chichester, London, 1975Google Scholar
  14. 14.
    Janson, S.: Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110(2), 177–245 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Jiřina, M.: Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8(83), 292–313 (1958)Google Scholar
  16. 16.
    Johnson, N.L., Kotz, S.: Urn models and their application. Wiley, New York, 1977Google Scholar
  17. 17.
    Kallenberg, O.: Foundations of modern probability. 2nd ed., Springer-Verlag, New York, 2002Google Scholar
  18. 18.
    Knuth, D.E.: The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 3nd ed., Addison-Wesley, Reading, Mass., 1997Google Scholar
  19. 19.
    Kotz, S., Mahmoud, H., Robert, P.: On generalized Pólya urn models. Statist. Probab. Lett. 49(2), 163–173 (2000)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Pemantle, R., Volkov, S.: Vertex-reinforced random walk on Z has finite range. Ann. Probab. 27(3), 1368–1388 (1999)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Pollard, H.: The completely monotonic character of the Mittag-Leffler function E a(-x). Bull. Amer. Math. Soc. 54, 1115–1116 (1948)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pólya, G.: Sur quelques points de la théorie des probabilités. Ann. Inst. Poincaré 1, 117–161 (1931)Google Scholar
  23. 23.
    Puyhaubert, V.: Modèles d'urnes et phénomènes de seuils en combinatoire analytique. Ph.D. thesis, I'École Polytechnique, 2005.Google Scholar
  24. 24.
    Smythe, R.T.: Central limit theorems for urn models. Stochastic Process. Appl. 65(1), 115–137 (1996)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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