Abstract
We investigate the pólya process, which underlies an urn of white and blue balls growing in real time. A partial differential equation governs the evolution of the process. For urns with (forward or backward) diagonal ball addition matrix the partial differential equation is amenable to asymptotic solution. In the case of forward diagonal we find a solution via the method of characteristics; the numbers of white and blue balls, when scaled appropriately, converge in distribution to independent Gamma random variables. The method of characteristics becomes a bit too involved for the backward diagonal process, except in degenerate cases, where we have Poisson behavior. In nondegenerate cases, limits characterized implicitly by their recursive sequence of moments are found, via matrix formulation involving a Leonard pair.
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References
Athreya K., Karlin S. (1968). Embedding of urn schemes into continuous time Markov branching process and related limit theorems. Annals of Mathematical Statistics 39, 1801–1817
Eggenberger F., pólya G. (1923). Über die Statistik verketetter Vorgäge. Zeitschrift für Angewandte Mathematik und Mechanik 1, 279–289
Freedman D. (1965). Bernard Friedman’s urn. Annals of Mathematical Statistics 36, 956–970
Friedman B. (1949). A simple urn model. Communications of Pure and Applied Mathematics 2, 59–70
Johnson N., Kotz S. (1977). Urn Models and Their Applications. Wiley, New York
Kotz S., Balakrishnan N. (1997). Advances in urn models during the past two decades. Advances in Combinatorial Methods and Applications to Probability and Statistics 49, 203–257
Kotz S., Mahmoud H., Robert P. (2000). On generalized pólya urn models. Statistics and Probability Letters 49, 163–173
Levine H. (1997). Partial Differential Equations. American Mathematical Society and International Press, Providence, Rhode Island
Mahmoud H. (2002). The size of random bucket trees via urn models. Acta Informatica 38, 813–838
Mahmoud H. (2003). Urn models and connections to random trees: A review. Journal of the Iranian Statistical Society 2, 53–114
Mahmoud H. (2004). Random sprouts as internet models and pólya processes. Acta Informatica 41, 1–18
Rosenblatt A. (1940). Sur le concept de contagion de M. G. pólya dans le calcul des probabilités. Proc. Acad. Nac. Cien. Exactas, Fis. Nat., Peru (Lima) 3, 186–204
Smiley M. (1965). Algebra of Matrices. Allyn and Bacon, Inc., Boston, Massachusetts
Terwilliger P. (2000). Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra and Applications 330, 149–203
Terwilliger P. (2002). Leonard pairs from 24 points of view. Rocky Mountain journal of Mathematics 32, 827–888
Terwilliger P. (2004). Leonard pairs and q-racah polynomials. Linear Algebra and its Applications 387, 235–276
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An erratum to this article can be found at http://dx.doi.org/10.1007/s10463-006-0093-1
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Balaji, S., Mahmoud, H.M. Exact and Limiting Distributions in Diagonal Pólya Processes. Ann Inst Stat Math 58, 171–185 (2006). https://doi.org/10.1007/s10463-005-0012-x
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DOI: https://doi.org/10.1007/s10463-005-0012-x