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Exactly solvable urn models under random replacement schemes and their applications

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 February 2023

Kiyoshi Inoue
Kiyoshi Inoue*
Affiliation:
Seikei University
*
*Postal address: Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musasino-shi, Tokyo, 180-8633, Japan. Email: kinoue@econ.seikei.ac.jp
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Abstract

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

Keywords

Urn modelrandom replacement schemedifferential equationswaiting timesenumerationprobability functionprobability generating functionEhrenfest urnbirthday problemcoupon collector’s problem

MSC classification

Primary: 60E05: Distributions
Secondary: 62E15: Exact distribution theory
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 835 - 854
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39, 1801–1817.CrossRefGoogle Scholar
Balaji, S. and Mahmoud, H. (2006). Exact and limiting distributions in diagonal Pólya processes. Ann. Inst. Statist. Math. 58, 171–185.CrossRefGoogle Scholar
Blom, G., Holst, L. and Sandell, D. (1994). Problems and Snapshots from the World of Probability. Springer, New York.CrossRefGoogle Scholar
Charalambides, Ch. A. (2002). Enumerative Combinatorics. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Charalambides, Ch. A. (2005). Combinatorial Methods in Discrete Distributions. John Wiley, New York.CrossRefGoogle Scholar
Chen, C. and Zhang, P. (2019). Characterizations of asymptotic distributions of continuous-time Polya processes. Commun. Statist. Theory Meth. 48, 5308–5321.CrossRefGoogle Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th edn. Cambridge University Press.CrossRefGoogle Scholar
Flajolet, P., Gabarro, J. and Pekari, H. (2005). Analytic urns. Ann. Prob. 33, 1200–1233.10.1214/009117905000000026CrossRefGoogle Scholar
Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Proc. 4th Colloq. Math. Comp. Sci.: Algorithms, Trees, Combinatorics and Probabilities, ed. P. Chassaing, pp. 59–118.Google Scholar
Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.CrossRefGoogle Scholar
Hwang, H.-K., Kuba, M. and Panholzer, A. (2007). Analysis of some exactly solvable diminishing urn models. In Proc. 19th Int. Conf. Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin.Google Scholar
Hwang, H.-K., Chern, H.-H. and Duh, G.-H. (2020). An asymptotic distribution theory for Eulerian recurrences with applications. To appear in Adv. Appl. Math. CrossRefGoogle Scholar
Inoue, K. and Aki, S. (2001). Pólya urn models under general replacement schemes. J. Japan Statist. Soc. 31, 193–205.CrossRefGoogle Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications. Wiley, New York.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley, New York.Google Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. Wiley, New York.CrossRefGoogle Scholar
Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. N. Balakrishnan. Birkhäuser, Boston, MA, pp. 203–257.CrossRefGoogle Scholar
Koutras, M. V. (1982). Non-central Stirling numbers and some applications. Discrete Math. 42, 73–89.CrossRefGoogle Scholar
Mahmoud, H. (2008). Pólya Urn Models. Chapman and Hall, Boca Raton, FL.CrossRefGoogle Scholar
Morcrette, B. and Mahmoud, H. (2012). Exactly solvable balanced tenable urns with random entries via the analytic methodology. In Proc. 23rd Int. Meeting Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, pp. 219–232.10.46298/dmtcs.2996CrossRefGoogle Scholar
Smythe, R. T. (1996). Central limit theorems for urn models. Stoch. Process. Appl. 65, 115–137.CrossRefGoogle Scholar