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Random additions in urns of integers

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  23 June 2021

Mackenzie Simper Open the ORCID record for Mackenzie Simper [Opens in a new window]
Mackenzie Simper*
Affiliation:
Stanford University
*
*Postal address: Department of Mathematics, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125. Email address: [email protected].
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Abstract

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

Keywords

urn modelexponential limit lawrecursive distributional equationcontraction method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 335 - 346
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abrams, A., Landau, H., Landau, Z., Pommersheim, J. and Zaslow, E. (2007). Random multiplication approaches uniform measure in finite groups. J. Theoret. Prob. 20, 107–118.10.1007/s10959-006-0051-0CrossRefGoogle Scholar
Bandyopadhyay, A. and Thacker, D. (2017). Pólya urn schemes with infinitely many colors. Bernoulli 23, 3243–3267.10.3150/16-BEJ844CrossRefGoogle Scholar
Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.10.1214/aos/1176345637CrossRefGoogle Scholar
Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1, 353–355.CrossRefGoogle Scholar
Janson, S. (2018). A.s. convergence for infinite colour Pólya urns associated with random walks. Preprint, .Google Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley & Sons, New York.Google Scholar
Knape, M. and Neininger, R. (2014). Pólya urns via the contraction method. Combinatorics Prob. Comput. 23, 1148–1186.10.1017/S0963548314000364CrossRefGoogle Scholar
Kotz, S. and Steutel, F. W. (1988). Note on a characterization of exponential distributions. Statist. Prob. Lett. 6, 201–203.10.1016/0167-7152(88)90120-4CrossRefGoogle Scholar
Kuba, M., Mahmoud, H. and Panholzer, A. (2013). Analysis of a generalized Friedman’s urn with multiple drawings. Discrete Appl. Math. 161, 2968–2984.10.1016/j.dam.2013.06.022CrossRefGoogle Scholar
Kuba, M. and Mahmoud, H. M. (2017). Two-color balanced affine urn models with multiple drawings. Adv. Appl. Math. 90, 1–26.10.1016/j.aam.2017.04.004CrossRefGoogle Scholar
Lasmar, N., Mailler, C. and Selmi, O. (2018). Multiple drawing multi-colour urns by stochastic approximation. J. Appl. Prob. 55, 254–281.10.1017/jpr.2018.16CrossRefGoogle Scholar
McKenzie Alexander, J., Skyrms, B. and Zabell, S. (2012). Inventing new signals. Dynamic Games Appl. 2, 129–145.10.1007/s13235-011-0027-2CrossRefGoogle Scholar
Mahmoud, H. M. (2009). Pólya Urn Models. CRC Press, Boca Raton, FL.Google Scholar
Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19, 498–524.10.1002/rsa.10010CrossRefGoogle Scholar
Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378–418.10.1214/aoap/1075828056CrossRefGoogle Scholar
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 1–79.10.1214/07-PS094CrossRefGoogle Scholar
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770–799.10.2307/1428133CrossRefGoogle Scholar
RÖsler, U. (1991). A limit theorem for “Quicksort.” RAIRO Inform. Théor. Appl. 25, 85–100.Google Scholar
Siegmund, D. and Yakir, B. (2005). An urn model of Diaconis. Ann. Prob. 33, 2036–2042.10.1214/009117905000000314CrossRefGoogle Scholar
Skyrms, B. (2010). Signals: Evolution, Learning, and Information. Oxford University Press.CrossRefGoogle Scholar
Thacker, D. (2015). Infinite color urn models. PhD thesis. Indian Statistical Institute, Dehli Centre.Google Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.10.1017/CBO9780511813658CrossRefGoogle Scholar