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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.

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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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