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DRAWING MULTISETS OF BALLS FROM TENABLE BALANCED LINEAR URNS

Published online by Cambridge University Press:  28 March 2013

Hosam M. Mahmoud
Hosam M. Mahmoud*
Affiliation:
The George Washington University, Washington, D.C. 20052, USA E-mail: hosam@gwu.edu
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Abstract

We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball addition according to the outcome. A nonsquare ball addition matrix of the form $\left( \matrix{a & b \cr c & d \cr e & f}\right)$ corresponds to such a scheme, in contrast to pólya urn models that possess a square ball addition matrix. We look into the case of constant row sum (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns: the nondegenerate and the degenerate. Via martingales, in the nondegenerate case one gets an asymptotic normal distribution for the number of balls of any color. In the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row sum.

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Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 2 , April 2013 , pp. 147 - 162
Copyright
Copyright © Cambridge University Press 2013

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References

1.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.CrossRefGoogle Scholar
2.Athreya, K. & Ney, P. (1972). Branching Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
3.Bagchi, A. & Pal, A. (1985). Asymptotic normality in the generalized Pólya-Eggenberger urn model with applications to computer data structures. SIAM Journal on Algebraic and Discrete Methods 6: 394–405.CrossRefGoogle Scholar
4.Bai, Z. & Hu, F. (1999). Asymptotic theorems for urn models with non-homogeneous generating matrices. Stochastic Processes and Their Applications 80: 87–101.CrossRefGoogle Scholar
5.Balaji, S. & Mahmoud, H. (2006). Exact and Limiting distributions in diagonal Pólya processes. The Annals of the Institute of Statistical Mathematics 58: 171–185.CrossRefGoogle Scholar
6.Balaji, S., Mahmoud, H., & Watanabe, O. (2006). Distributions in the Ehrenfest process. Statistics and Probability Letters 76: 666–674.CrossRefGoogle Scholar
7.Bernstein, S. (1940). Sur un problème du schéma des urnes à composition variable. Comptes Rendus (Doklady) de l’ Academic des Sciences de l’ URSS 28: 5–7.Google Scholar
8.Chen, M. & Wei, C. (2005). A new urn model. Journal of Applied Probability 42: 964–976.CrossRefGoogle Scholar
9.Eggenberger, F. & Pólya, G. (1923). Über die statistik verketetter vorgäge. Zeitschrift für Angewandte Mathematik und Mechanik 1: 279–289.CrossRefGoogle Scholar
10.Flajolet, P., Dumas, P., & Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Philippe Chassaing AG, (Ed) Discrete Mathematics and Computer Science Proceedings, 59–118.CrossRefGoogle Scholar
11.Flajolet, P., Gabarró, J., & Pekari, H. (2005). Analytic urns. The Annals of Probability 33: 1200–1233.CrossRefGoogle Scholar
12.Freedman, D. (1965). Bernard Friedman's urn. Annals of Mathematical Statistics 36: 956–970.CrossRefGoogle Scholar
13.Friedman, B. (1949). A simple urn model. Communications of Pure and Applied Mathematics 2: 59–70.CrossRefGoogle Scholar
14.Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya-Eggenberger urn model. Statistics and Probability Letters 8: 225–228.CrossRefGoogle Scholar
15.Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya Urn. Annals of Probability 21: 1624–1639.CrossRefGoogle Scholar
16.Hall, P. & Heyde, C. (1980). Martingale limit theory and its applications. New York: Academic Press.Google Scholar
17.Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Processes and Applications 110: 177–245.CrossRefGoogle Scholar
18.Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures and Algorithms 26: 69–83.CrossRefGoogle Scholar
19.Johnson, N. & Kotz, S. (1977). Urn Models and their applications. New York: Wiley.Google Scholar
20.Johnson, N., Kotz, S., & Mahmoud, H. (2004). Pólya-type urn models with multiple drawings. Journal of the Iranian Statistical Society 3: 165–173.Google Scholar
21.Kotz, S. & Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Balakrishman, N. (ed.), Advances in combinatorial methods and applications to probability and statistics, vol 49, Boston: Birkhäuser, pp. 203–257.CrossRefGoogle Scholar
22.Kotz, S., Mahmoud, H., & Robert, P. (2000). On generalized pólya urn models. Statistics and Probability Letters 49: 163–173.CrossRefGoogle Scholar
23.Mahmoud, H. (1998). On rotations in fringe-balanced binary trees. Information Processing Letters 65: 41–46.CrossRefGoogle Scholar
24.Mahmoud, H. (2003). Pólya urn models and connections to random trees: A review. Journal of the Iranian Statistical Society 2: 53–114.Google Scholar
25.Mahmoud, H. & Smythe, R. (1992). Asymptotic joint normality of outdegrees of nodes in random recursive trees. Random Structures and Algorithms 3: 255–266.CrossRefGoogle Scholar
26.Mahmoud, H. & Smythe, R. (1995). Probabilistic analysis of bucket recursive trees. Theoretical Computer Science 144: 221–249.CrossRefGoogle Scholar
27.Mahmoud, H., Smythe, R., & Szymański, J. (1993). On the structure of plane-oriented recursive trees and their branches. Random Structures and Algorithms 4: 151–176.CrossRefGoogle Scholar
28.Mahmoud, H. & Tsukiji, T. (2000). A limit law for outputs in random recursive circuits. Algorithmica 31: 403–412.Google Scholar
29.Pemantle, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations. Annals of Probability 18: 698–712.CrossRefGoogle Scholar
30.Pemantle, R. (2007). A survey of random processes with reinforcement. Probability Surveys 4: 1–79.CrossRefGoogle Scholar
31.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.Google Scholar
32.Savkevich, V. (1940). Sur le schéma des urnes à composition variable. Comptes Rendus (Doklady) de l’ Academic des Sciences de l’ URSS 28: 8–12.Google Scholar
33.Smythe, R. (1996). Central limit theorems for urn models. Stochastic Processes and Their Applications 65: 115–137.CrossRefGoogle Scholar