Bernoulli

  • Bernoulli
  • Volume 23, Number 4B (2017), 3243-3267.

Pólya urn schemes with infinitely many colors

Antar Bandyopadhyay and Debleena Thacker

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Abstract

In this work, we introduce a class of balanced urn schemes with infinitely many colors indexed by ${\mathbb{Z} }^{d}$, where the replacement schemes are given by the transition matrices associated with bounded increment random walks. We show that the color of the $n$th selected ball follows a Gaussian distribution on ${\mathbb{R} }^{d}$ after ${\mathcal{O} }(\log n)$ centering and ${\mathcal{O} }(\sqrt{\log n})$ scaling irrespective of whether the underlying walk is null recurrent or transient. We also provide finer asymptotic similar to local limit theorems for the expected configuration of the urn. The proofs are based on a novel representation of the color of the $n$th selected ball as “slowed down” version of the underlying random walk.

Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3243-3267.

Dates
Received: July 2015
Revised: February 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1495505092

Digital Object Identifier
doi:10.3150/16-BEJ844

Mathematical Reviews number (MathSciNet)
MR3654806

Zentralblatt MATH identifier
06778286

Keywords
central limit theorem infinite color urn local limit theorem random walk reinforcement processes urn models

Citation

Bandyopadhyay, Antar; Thacker, Debleena. Pólya urn schemes with infinitely many colors. Bernoulli 23 (2017), no. 4B, 3243--3267. doi:10.3150/16-BEJ844. https://projecteuclid.org/euclid.bj/1495505092


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References

  • [1] Athreya, K.B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Stat. 39 1801–1817.
  • [2] 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. Algebr. Discrete Methods 6 394–405.
  • [3] Bai, Z.-D. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Probab. 15 914–940.
  • [4] Bhattacharya, R.N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley Series in Probability and Mathematical Statistics. New York–London–Sydney: John Wiley & Sons.
  • [5] Biggins, J.D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
  • [6] Biggins, J.D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
  • [7] Biggins, J.D. (1998). Lindley-type equations in the branching random walk. Stochastic Process. Appl. 75 105–133.
  • [8] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. A Wiley-Interscience Publication.
  • [9] Blackwell, D. (1973). Discreteness of Ferguson selections. Ann. Statist. 1 356–358.
  • [10] Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
  • [11] Bose, A., Dasgupta, A. and Maulik, K. (2009). Multicolor urn models with reducible replacement matrices. Bernoulli 15 279–295.
  • [12] Bose, A., Dasgupta, A. and Maulik, K. (2009). Strong laws for balanced triangular urns. J. Appl. Probab. 46 571–584.
  • [13] Chen, M.-R., Hsiau, S.-R. and Yang, T.-H. (2014). A new two-urn model. J. Appl. Probab. 51 590–597.
  • [14] Chen, M.-R. and Kuba, M. (2013). On generalized Pólya urn models. J. Appl. Probab. 50 1169–1186.
  • [15] Collevecchio, A., Cotar, C. and LiCalzi, M. (2013). On a preferential attachment and generalized Pólya’s urn model. Ann. Appl. Probab. 23 1219–1253.
  • [16] Conway, J.B. (1978). Functions of One Complex Variable, 2nd ed. Graduate Texts in Mathematics 11. New York–Berlin: Springer.
  • [17] Cotar, C. and Limic, V. (2009). Attraction time for strongly reinforced walks. Ann. Appl. Probab. 19 1972–2007.
  • [18] Crane, E., Georgiou, N., Volkov, S., Wade, A.R. and Waters, R.J. (2011). The simple harmonic urn. Ann. Probab. 39 2119–2177.
  • [19] Dasgupta, A. and Maulik, K. (2011). Strong laws for urn models with balanced replacement matrices. Electron. J. Probab. 16 1723–1749.
  • [20] Davis, B. (1990). Reinforced random walk. Probab. Theory Related Fields 84 203–229.
  • [21] Devroye, L. (1988). Applications of the theory of records in the study of random trees. Acta Inform. 26 123–130.
  • [22] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
  • [23] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • [24] Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities. Discrete Math. Theor. Comput. Sci. Proc., AG 59–118. Nancy: Assoc. Discrete Math. Theor. Comput. Sci.
  • [25] Freedman, D.A. (1965). Bernard Friedman’s urn. Ann. Math. Stat. 36 956–970.
  • [26] Friedman, B. (1949). A simple urn model. Comm. Pure Appl. Math. 2 59–70.
  • [27] Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn. J. Appl. Probab. 34 426–435.
  • [28] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.
  • [29] Janson, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417–452.
  • [30] Laruelle, S. and Pagès, G. (2013). Randomized urn models revisited using stochastic approximation. Ann. Appl. Probab. 23 1409–1436.
  • [31] Launay, M. and Limic, V. (2012). Generalized interacting urn models. Preprint. Available at arXiv:1207.5635.
  • [32] Lawler, G.F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge: Cambridge Univ. Press.
  • [33] Limic, V. (2003). Attracting edge property for a class of reinforced random walks. Ann. Probab. 31 1615–1654.
  • [34] Limic, V. and Tarrès, P. (2007). Attracting edge and strongly edge reinforced walks. Ann. Probab. 35 1783–1806.
  • [35] Pemantle, R. (1990). A time-dependent version of Pólya’s urn. J. Theoret. Probab. 3 627–637.
  • [36] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
  • [37] Petrov, V.V. (1975). Sums of Independent Random Variables. Ergebnisse der Mathematik und Ihrer Grenzgebiete 82. New York–Heidelberg: Springer. Translated from the Russian by A. A. Brown.
  • [38] Pólya, G. (1930). Sur quelques points de la théorie des probabilités. Ann. Inst. Henri Poincaré 1 117–161.
  • [39] Smythe, R.T. and Mahmoud, H.M. (1994). A survey of recursive trees. Teor. Ĭmovīr. Mat. Stat. 51 1–29.
  • [40] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.