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Large Deviations Principle for Occupancy Problems with Colored Balls

Part of: Limit theorems Combinatorics

Published online by Cambridge University Press:  14 July 2016

Paul Dupuis ,
Carl Nuzman  and
Phil Whiting
Paul Dupuis*
Affiliation:
Brown University
Carl Nuzman*
Affiliation:
Alcatel-Lucent
Phil Whiting*
Affiliation:
Alcatel-Lucent
*
∗ Research supported in part by the National Science Foundation (NSF-DMS-0306070 and NSF-DMS-0404806) and the Army Research Office (DAAD19-02-1-0425).
∗ Research supported in part by the National Science Foundation (NSF-DMS-0306070 and NSF-DMS-0404806) and the Army Research Office (DAAD19-02-1-0425).
∗ Research supported in part by the National Science Foundation (NSF-DMS-0306070 and NSF-DMS-0404806) and the Army Research Office (DAAD19-02-1-0425).
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Abstract

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A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

Keywords

Occupancy modelcoloring processMaxwell-Boltzmann statisticssample path large deviations principlecircuit-switched networkpopulation estimationoptical packet switching

MSC classification

Primary: 60F10: Large deviations
Secondary: 05A16: Asymptotic enumeration
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 115 - 141
Copyright
Copyright © Applied Probability Trust 2007 

References

Bailey, N. T. J. (1951). On estimating the size of mobile populations from recapture data. Biometrika 38, 293–306.CrossRefGoogle Scholar
Boucheron, S., Gamboa, F. and Leonard, C. (2002). Bins and balls: large deviations of the empirical occupancy process. Ann. Appl. Prob. 2, 1–30.Google Scholar
Chao, A. (2001). An overview of closed capture–recapture models. J. Agricultural Biol. Environ. Statist. 6, 158–175.CrossRefGoogle Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to Large Deviations. John Wiley, New York.CrossRefGoogle Scholar
Dupuis, P., Ellis, R. S. and Weiss, A. (1991). Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Prob. 19, 1280–1297.CrossRefGoogle Scholar
Dupuis, P., Nuzman, C. and Whiting, P. (2003). Occupancy models and circuit switched networks with blocking. In Proc. 41st Annual Allerton Conf. Commun. Control Comput. (September 2003), University of Illinois Press, Champaign, IL.Google Scholar
Dupuis, P., Nuzman, C. and Whiting, P. (2004). Large deviation asymptotics for occupancy problems. Ann. Prob. 32, 2765–2818.CrossRefGoogle Scholar
Eramo, V., Listanti, M., Nuzman, C. and Whiting, P. (2002). Optical switch dimensioning and the classical occupancy problem. Internat. J. Commun. 15, 127–141.CrossRefGoogle Scholar
Finkelstein, M., Tucker, H. G. and Veeh, J. A. (1998). Confidence intervals for the number of unseen types. Statist. Prob. Lett. 37, 423–430.CrossRefGoogle Scholar
Graham, C. and O'Connell, N. (2002). Large deviations at equilibrium for a large star-shaped loss network. Ann. Appl. Prob. 2, 1807–1856.Google Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
Kelly, F. P. and Ziedins, I. (1989). Blocking in star networks. Adv. Appl. Prob. 21, 804–830.Google Scholar
Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. Balakrishnan, N., Birkhäuser, Boston, MA, pp. 203–257.CrossRefGoogle Scholar
Seber, G. A. F. (1982). The Estimation of Animal Abundance and Related Parameters, 2nd edn. Macmillan, New York.Google Scholar
Vander Wiel, S. A. and Votta, L. G. (1993). Assessing software designs using capture-recapture methods. IEEE Trans. Software Eng. 19, 1045–1054.CrossRefGoogle Scholar
Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. AT&T Tech. J. 64, 1807–1856.Google Scholar