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Large-Deviation Approximations for General Occupancy Models

Published online by Cambridge University Press:  01 May 2008

JIM X. ZHANG  and
PAUL DUPUIS
JIM X. ZHANG
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (e-mail: jim.zhang@ubs.com)
PAUL DUPUIS
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (e-mail: jim.zhang@ubs.com)
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Abstract

We obtain large-deviation approximations for the empirical distribution for a general family of occupancy problems. In the general setting, balls are allowed to fall in a given urn depending on the urn's contents prior to the throw. We discuss a parametric family of statistical models that includes Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics as special cases. A process-level large-deviation analysis is conducted and the rate function for the original problem is then characterized, via the contraction principle, by the solution to a calculus of variations problem. The solution to this variational problem is shown to coincide with that of a simple finite-dimensional minimization problem. As a consequence, the large-deviation approximations and related qualitative information are available in more-or-less explicit form.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 437 - 470
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Billingsley, P. (1968) Convergence of Probability Measures, Wiley, New York.Google Scholar
[2]Charalambides, A. (1997) A unified derivation of occupancy and sequential occupancy distributions. In Advances in Combinatorial Methods and Applications to Probability and Statistics (Balakrishnan, N., ed.), pp. 259–273.CrossRefGoogle Scholar
[3]Dupuis, P. and Ellis, R. (1997) A Weak Convergence Approach to Large Deviations, Wiley–Interscience, New York.CrossRefGoogle Scholar
[4]Dupuis, P., Nuzman, C. and Whiting, P. (2004) Large deviations asymptotics for occupancy problems. Ann. Probab. 32 2765–2818.CrossRefGoogle Scholar
[5]Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, Wiley, New York.Google Scholar
[6]Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York.Google Scholar
[7]Fleming, W. H. and Rishel, R. W. (1982) Deterministic and Stochastic Optimal Control, Springer.CrossRefGoogle Scholar
[8]Holst, L. (1986) On the coupon collectors and other urn problems. Internat. Statist. Review 54 15–27.CrossRefGoogle Scholar
[9]Johnson, N. L. and Kotz, S. (1977) Urn Models and their Applications, Wiley.Google Scholar
[10]Rockafellar, R. T. (1970) Convex Analysis, Princeton University Press.CrossRefGoogle Scholar
[11]Ross, S. (2002) Introduction to Probability Models, 8th edn, Academic Press.Google Scholar
[12]Zhang, J. X., Nuzman, C. and Whiting, P. (2005) Importance sampling in a general urn occupancy model. Preprint.Google Scholar