Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T01:18:41.664Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong convergence of infinite color balanced urns under uniform ergodicity

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 September 2020

Antar Bandyopadhyay ,
Svante Janson  and
Debleena Thacker
Antar Bandyopadhyay*
Affiliation:
Indian Statistical Institute, Delhi and Kolkata
Svante Janson*
Affiliation:
Uppsala University
Debleena Thacker*
Affiliation:
Uppsala University
*
*Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi Centre, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India. Email address: antar@isid.ac.in
**Postal address: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden. Email address: svante@math.uu.se
**Postal address: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden. Email address: svante@math.uu.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

Keywords

Almost sure convergencebranching Markov chaininfinite color urnrandom recursive treereinforcement processesuniform ergodicityurn models

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 853 - 865
Check for updates
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. (1985). Exchangeability and related topics. In École d’Été de Probabilités de Saint-Flour, XIII—1983 (Lecture Notes Math. 1117), pp. 1–198. Springer, Berlin.CrossRefGoogle Scholar
Athreya, K. B. andKarlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39, 1801–1817.CrossRefGoogle Scholar
Bai, Z.-D. andHu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Prob. 15 (1B), 914–940.CrossRefGoogle Scholar
Bandyopadhyay, A. andThacker, D. (2014). Rate of convergence and large deviation for the infinite color Pólya urn schemes. Statist. Prob. Lett. 92, 232–240.CrossRefGoogle Scholar
Bandyopadhyay, A. andThacker, D. (2016). A new approach to Pólya urn schemes and its infinite color generalization. Available at arXiv:1606.05317.Google Scholar
Bandyopadhyay, A. andThacker, D. (2017). Pólya urn schemes with infinitely many colors. Bernoulli 23 (4B), 3243–3267.CrossRefGoogle Scholar
Blackwell, D. andMacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1, 353–355.CrossRefGoogle Scholar
Bose, A., Dasgupta, A. andMaulik, K. (2009). Multicolor urn models with reducible replacement matrices. Bernoulli 15 (1), 279–295.CrossRefGoogle Scholar
Bose, A., Dasgupta, A. andMaulik, K. (2009). Strong laws for balanced triangular urns. J. Appl. Prob. 46 (2), 571–584.CrossRefGoogle Scholar
Chen, M.-R. andKuba, M. (2013). On generalized Pólya urn models. J. Appl. Prob. 50 (4), 1169–1186.CrossRefGoogle Scholar
Chen, M.-R., Hsiau, S.-R. andYang, T.-H. (2014). A new two-urn model. J. Appl. Prob. 51 (2), 590–597.CrossRefGoogle Scholar
Dasgupta, A. andMaulik, K. (2011). Strong laws for urn models with balanced replacement matrices. Electron. J. Prob. 16 (63), 1723–1749.CrossRefGoogle Scholar
Donnelly, P. andKurtz, T. G. (1996). The asymptotic behavior of an urn model arising in population genetics. Stoch. Process. Appl. 64 (1), 1–16.CrossRefGoogle Scholar
Drmota, M. (2009). Random Trees. Springer, Vienna.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. John Wiley, New York, London and Sydney.Google Scholar
Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.CrossRefGoogle Scholar
Flajolet, P., Dumas, P. andPuyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Fourth Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proc. AG, 59–118.Google Scholar
Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn. J. Appl. Prob. 34 (2), 426–435.CrossRefGoogle Scholar
Häggström, O. (2005). On the central limit theorem for geometrically ergodic Markov chains. Prob. Theory Relat. Fields 132 (1), 74–82.CrossRefGoogle Scholar
Hirth, U. M. (1997). From GEM back to Dirichlet via Hoppe’s urn. Combin. Probab. Comput. 6 (2), 185–195.CrossRefGoogle Scholar
Holst, L. (2008). The number of two consecutive successes in a Hoppe–Pólya urn. J. Appl. Prob. 45 (3), 901–906.CrossRefGoogle Scholar
Hoppe, F. M. (1984). Pólya-like urns and the Ewens’ sampling formula. J. Math. Biol. 20 (1), 91–94.CrossRefGoogle Scholar
Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25 (2), 123–159.CrossRefGoogle Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110 (2), 177–245.CrossRefGoogle Scholar
Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134 (3), 417–452.CrossRefGoogle Scholar
Janson, S. (2018). A.s. convergence for infinite colour Pólya urns associated with random walks. Available at arXiv:1803.04207.Google Scholar
Janson, S. (2019). Random replacements in Pólya urns with infinitely many colours. Electron. Commun. Prob. 24, 23, 1–11.CrossRefGoogle Scholar
Laruelle, S. andPagès, G. (2013). Randomized urn models revisited using stochastic approximation. Ann. Appl. Prob. 23 (4), 1409–1436.CrossRefGoogle Scholar
Levin, D. A., Peres, Y. andWilmer, E. L. (2009). Markov Chains and Mixing Times. American Mathematical Society, Providence, RI.Google Scholar
Lyons, R. (1988). Strong laws of large numbers for weakly correlated random variables. Michigan Math. J. 35 (3), 353–359.Google Scholar
Mailler, C. andMarckert, J.-F. (2017). Measure-valued Pólya urn processes. Electron. J. Prob. 22, 26, 1–33.CrossRefGoogle Scholar
Meyn, S. andTweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 1–79.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.Google Scholar
Pólya, G. (1930). Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1 (2), 117–161.Google Scholar
Seneta, E. (2006). Non-negative Matrices and Markov Chains (SpringerSeries in Statistics). Springer, New York.Google Scholar
Thacker, D. (2015). Infinite color urn models. Doctoral thesis, Indian Statistical Institute, Delhi Centre.Google Scholar
Thörnblad, E. (2016). The dominating colour of an infinite Pólya urn model. J. Appl. Prob. 53 (3), 914–924.CrossRefGoogle Scholar