The Annals of Applied Probability

The Bernoulli sieve revisited

Alexander V. Gnedin, Alexander M. Iksanov, Pavlo Negadajlov, and Uwe Rösler

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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.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 4 (2009), 1634-1655.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700630

Digital Object Identifier
doi:10.1214/08-AAP592

Mathematical Reviews number (MathSciNet)
MR2538083

Zentralblatt MATH identifier
1178.60019

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Keywords
Occupancy residual allocation model distributional recursion regenerative composition

Citation

Gnedin, Alexander V.; Iksanov, Alexander M.; Negadajlov, Pavlo; Rösler, Uwe. The Bernoulli sieve revisited. Ann. Appl. Probab. 19 (2009), no. 4, 1634--1655. doi:10.1214/08-AAP592. https://projecteuclid.org/euclid.aoap/1248700630


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