An almost sure limit theorem for mixtures of domains in random allocation

Peter Becker-Kern

doi:10.1556/sscmath.2007.1024

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 44: Issue 3

An almost sure limit theorem for mixtures of domains in random allocation

Author:

Peter Becker-Kern

Peter Becker-Kern Universität Dortmund Fachbereich Mathematik 44221 Dortmund Germany

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Pages:: 331–354

Online Publication Date:: 11 May 2007

Publication Date:: 01 Sep 2007

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/sscmath.2007.1024

Keywords:: Primary 60F15; Secondary 60F05; Almost sure limit theorem; random allocation; logarithmic weights; transfer theorem; mixtures of probability measures; empty cell test; occupancy problem

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The problem of random allocation is that of placing n balls independently with equal probability to N boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov [10] and show that asymptotic normality dominates.

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Studia Scientiarum Mathematicarum Hungarica

Print ISSN:: 0081-6906

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János PACH (Rényi Institute of Mathematics)
Péter Pál PACH (BME, Budapest)
Andrew SUK (University of California, San Diego)
Zoltán SZABÓ (Princeton University)
Martin TANCER (Charles University, Prague)
Gábor TARDOS (Rényi Institute of Mathematics)
Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
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