Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T04:23:04.223Z Has data issue: false hasContentIssue false

Mean and variance of balanced Pólya urns

Published online by Cambridge University Press:  03 December 2020

Svante Janson*
Affiliation:
Uppsala University
*
*Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email: svante.janson@math.uu.se

Abstract

It is well-known that in a small Pólya urn, i.e., an urn where the second largest real part of an eigenvalue is at most half the largest eigenvalue, the distribution of the numbers of balls of different colours in the urn is asymptotically normal under weak additional conditions. We consider the balanced case, and then give asymptotics of the mean and the covariance matrix, showing that after appropriate normalization, the mean and covariance matrix converge to the mean and covariance matrix of the limiting normal distribution.

Type
Original Article
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Prob. 6, 325331.10.1214/aop/1176995577CrossRefGoogle Scholar
Aldous, D. J., Flannery, B. and Palacios, J. L. (1988). Two applications of urn processes—the fringe analysis of search trees and the simulation of quasi-stationary distributions of Markov chains. Prob. Eng. Inf. Sci. 2, 293307.CrossRefGoogle Scholar
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, 18011817.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Bagchi, A. and Pal, A. K. (1985). Asymptotic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Meth. 6, 394405.CrossRefGoogle Scholar
Bai, Z.-D. and Hu, F. (1999). Asymptotic theorems for urn models with nonhomogeneous generating matrices. Stoch. Process. Appl. 80, 87101.CrossRefGoogle Scholar
Bai, Z.-D. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Prob. 15, 914940.CrossRefGoogle Scholar
Bai, Z.-D., Hu, F. and Rosenberger, W. F. (2002). Asymptotic properties of adaptive designs for clinical trials with delayed response. Ann. Statist. 30, 122139.CrossRefGoogle Scholar
Bernstein, S. N. (1940). Nouvelles applications des grandeurs aléatoires presqu’indépendantes. Izv. Akad. Nauk SSSR Ser. Mat. 4, 137150 (in Russian).Google Scholar
Bernstein, S. N. (1940). Sur un problème du schéma des urnes à composition variable. C. R. Acad. Sci. URSS 28, 57.Google Scholar
Devroye, L. (1991). Limit laws for local counters in random binary search trees. Random Structures Algorithms 2, 303315.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T. (1958). Linear Operators. Part I: General Theory. Interscience Publishers, Inc., New York.Google Scholar
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech. 3, 279289.CrossRefGoogle Scholar
Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Prob. 33, 12001233.10.1214/009117905000000026CrossRefGoogle Scholar
Holmgren, C. and Janson, S. (2015). Asymptotic distribution of two-protected nodes in ternary search trees. Electron. J. Prob. 20, 120.10.1214/EJP.v20-3577CrossRefGoogle Scholar
Holmgren, C., Janson, S. and Šileikis, M. (2017). Multivariate normal limit laws for the numbers of fringe subtrees in m-ary search trees and preferential attachment trees. Electron. J. Combin. 24, Paper 2.51, 49 pp.CrossRefGoogle Scholar
Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. Wiley-Interscience, Hoboken, NJ.CrossRefGoogle Scholar
Hu, F. and Zhang, L.-X. (2004). Asymptotic normality of urn models for clinical trials with delayed response. Bernoulli 10, 447463.CrossRefGoogle Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177245.CrossRefGoogle Scholar
Janson, S. (2005). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417452.CrossRefGoogle Scholar
Janson, S. and Pouyanne, N. (2018). Moment convergence of balanced Pólya processes. Electron. J. Prob. 23, paper no. 34, 13 pp.CrossRefGoogle Scholar
Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8, 292313.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. Wiley, New York.Google Scholar
Mahmoud, H. M. (2009). Pólya Urn Models. CRC Press, Boca Raton, FL.Google Scholar
Mahmoud, H. M. and Ward, M. D. (2012). Asymptotic distribution of two-protected nodes in random binary search trees. Appl. Math. Lett. 25, 22182222.CrossRefGoogle Scholar
Markov, A. A. (1917). Sur quelques formules limites du calcul des probabilités. Bull. Acad. Imp. Sci. 11, 177186 (in Russian).Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. Cambridge University Press. Also available as NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov.Google Scholar
Pólya, G. (1930). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1, 117161.Google Scholar
Pouyanne, N. (2008). An algebraic approach to Pólya processes. Ann. Inst. H. Poincaré Prob. Statist. 44, 293323.CrossRefGoogle Scholar
Savkevitch, V. (1940). Sur le schéma des urnes à composition variable. C. R. Acad. Sci. URSS 28, 812.Google Scholar
Zhang, L.-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Prob. 16, 340369.CrossRefGoogle Scholar