Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T22:16:19.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of the overflow in urn models

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  11 July 2022

Raul Gouet ,
Paweł Hitczenko Open the ORCID record for Paweł Hitczenko [Opens in a new window]  and
Jacek Wesołowski
Raul Gouet*
Affiliation:
Universidad de Chile
Paweł Hitczenko*
Affiliation:
National Science Foundation and Drexel University
Jacek Wesołowski*
Affiliation:
Warsaw University of Technology
*
*Postal address: Departamento de Ingenieria Matemática and CMM (IRL 2807, CNRS), Universidad de Chile, Beauchef 851, 8370456 Santiago, Chile. Email: [email protected]
**Postal address: Division of Mathematical Sciences, National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314, USA.
****Postal address: Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a finite or infinite collection of urns, each with capacity r, and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain r balls. When $r=1$ , this is the number of balls landing in non-empty urns, which has been studied in the past. Our aim here is to use martingale methods to study the asymptotics of the overflow in the general situation, i.e. for arbitrary r. In particular, we provide sufficient conditions for both Poissonian and normal asymptotics.

Keywords

Urn modeloccupancy problemrandom allocationweak limit theorem

MSC classification

Primary: 60F05: Central limit and other weak theorems 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 797 - 824
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Arratia, R., Garibaldi, S. and Kilian, J. (2016). Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process. Random Structures Algorithms 48, 480–502.10.1002/rsa.20591CrossRefGoogle Scholar
Barbour, A. Gnedin, A. (2009). Small counts in the infinite occupancy scheme. Electron. J. Prob. 14, 13, 365–384.10.1214/EJP.v14-608CrossRefGoogle Scholar
Beśka, M., Kłopotowski, A. and Słomiński, L. (1982). Limit theorems for random sums of dependent d-dimensional random vectors. Z. Wahrscheinlichkeitsth. 61, 43–57.10.1007/BF00537224CrossRefGoogle Scholar
Bobecka, K., Hitczenko, P., López-Blázquez, F., Rempała, G. and Wesołowski, J. (2013). Asymptotic normality through factorial cumulants and partition identities. Combinatorics Prob. Comput. 22, 213–240.10.1017/S0963548312000545CrossRefGoogle ScholarPubMed
Chao, A. and Chiu, C.-H. (2016). Species richness: Estimation and comparison. In Wiley StatsRef: Statistics Reference Online (eds N. Balakrishnan, T. Colton, B. Everitt, W. Piegorsch, F. Ruggeri and J. L. Teugels), John Wiley, London.Google Scholar
Dupuis, P., Nuzman, C. and Whiting, P. (2004). Large deviation asymptotics for occupancy problems. Ann. Prob. 32, 2765–2818.10.1214/009117904000000135CrossRefGoogle Scholar
Gnedin, A., Hansen, B. and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws. Prob. Surv. 4, 146–171.10.1214/07-PS092CrossRefGoogle Scholar
Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94.Google Scholar
Hwang, H. K. and Janson, S. (2008). Local limit theorems for finite and infinite urn models. Ann. Prob. 36, 992–1022.10.1214/07-AOP350CrossRefGoogle Scholar
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286–295.10.1214/aos/1176346079CrossRefGoogle Scholar
Jogdeo, K and Patil, G. P. (1975). Probability inequalities for certain multivariate discrete distribution. Sankhyā B 37, 158–164.Google Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application . Wiley Series in Probability and Mathematical Statistics. John Wiley, New York.Google Scholar
Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17, 373–401.Google Scholar
Knuth, D. E. (1998). The Art of Computer Programming, Vol. 2, 3rd edn. Addison-Wesley, Reading, MA.Google Scholar
Knuth, D. E. (1998). The Art of Computer Programming, Vol. 3, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar
Kolchin, V. F., Sevastyanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. Winston & Sons, Washington.Google Scholar
Kotz, S. and Balakrishnan, N. (1977). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. N. Balkrishnan. Birkhäuser, Boston, MA, pp. 203–257.Google Scholar
Mahmoud, H. M. (2009). Pólya Urn Models. CRC Press, Boca Raton, FL.Google Scholar
Mallows, C. L. (1968). An inequality involving multinomial probabilities. Biometrika 55, 422–424.10.1093/biomet/55.2.422CrossRefGoogle Scholar
Mikhailov, V. G. (1981). The central limit theorem for the scheme of independent placements of particles among cells. Trudy Steklov Math. Inst. (MIAN) 157, 138–152.Google Scholar
Monahan, J. F. (1987). An alternative method for computing the overflow probabilities. Commun. Statist. Theory Meth. 16, 3355–3357.10.1080/03610928708829575CrossRefGoogle Scholar
National Institute of Standards and Technology (2020). NIST Digital Library of Mathematical Functions, Release 1.0.28. (2020). Available at: https://dlmf.nist.gov.Google Scholar
Ramakrishna, M. V. (1987). Computing the probability of hash table/urn overflow. Commun. Statist. Theory Meth. 16, 3343–3353.10.1080/03610928708829574CrossRefGoogle Scholar