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Theory of Probability and Mathematical Statistics

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On the asymptotic behavior of a sequence of random variables of interest in the classical occupancy problem

Authors: Rita Giuliano and Claudio Macci
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 87 (2012).
Journal: Theor. Probability and Math. Statist. 87 (2013), 31-40
MSC (2010): Primary 60F10, 60C05
Published electronically: March 21, 2014
MathSciNet review: 3241444
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Abstract: In the classical occupancy problem one puts balls in $ n$ boxes, and each ball is independently assigned to any fixed box with probability $ \frac {1}{n}$. It is well known that, if we consider the random number $ T_n$ of balls required to have all the $ n$ boxes filled with at least one ball, the sequence $ \{T_n/(n\log n)\colon n\geq 2\}$ converges to 1 in probability. Here we present the large deviation principle associated to this convergence. We also discuss the use of the Gärtner Ellis Theorem for the proof of some parts of this large deviation principle.

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Additional Information

Rita Giuliano
Affiliation: Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy

Claudio Macci
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy

Keywords: Large deviation principle, coupon collector's problem, triangular array, Poisson approximation
Received by editor(s): December 13, 2011
Published electronically: March 21, 2014
Additional Notes: The financial support of the Research Grant PRIN 2008 Probability and Finance is gratefully acknowledged
The authors thank Francesco Pasquale for useful comments on the proof of \eqref{eq:UB} for $F∈\mathcal{C}_{2}$
Article copyright: © Copyright 2014 American Mathematical Society