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August 2009 The Bernoulli sieve revisited
Alexander V. Gnedin, Alexander M. Iksanov, Pavlo Negadajlov, Uwe Rösler
Ann. Appl. Probab. 19(4): 1634-1655 (August 2009). DOI: 10.1214/08-AAP592

Abstract

We consider an occupancy scheme in which “balls” are identified with n points sampled from the standard exponential distribution, while the role of “boxes” is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index Kn* of the last occupied box, the number Kn of occupied boxes, the number Kn, 0 of empty boxes whose index is at most Kn*, the index Wn of the first empty box and the number of balls Zn in the last occupied box. It is shown that the limiting distribution of properly scaled and centered Kn* coincides with that of the number of renewals not exceeding logn. A similar result is shown for Kn and Wn under a side condition that prevents occurrence of very small boxes. The condition also ensures that Kn, 0 converges in distribution. Limiting results for Zn are established under an assumption of regular variation.

Citation

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Alexander V. Gnedin. Alexander M. Iksanov. Pavlo Negadajlov. Uwe Rösler. "The Bernoulli sieve revisited." Ann. Appl. Probab. 19 (4) 1634 - 1655, August 2009. https://doi.org/10.1214/08-AAP592

Information

Published: August 2009
First available in Project Euclid: 27 July 2009

zbMATH: 1178.60019
MathSciNet: MR2538083
Digital Object Identifier: 10.1214/08-AAP592

Subjects:
Primary: 60F05
Secondary: 60C05

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 4 • August 2009
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