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Asymptotics of the overflow in urn models

Part of: Limit theorems Special processes

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: rgouet@dim.uchile.cl
**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: j.wesolowski@mini.pw.edu.pl
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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

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 797 - 824
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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