Electronic Journal of Probability

Regenerative compositions in the case of slow variation: A renewal theory approach

Alexander Gnedin and Alexander Iksanov

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A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the Lévy measure of the subordinator has a property of slow variation at $0$. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of Lévy measure at $\infty$. Similar results are also derived for the number of singleton blocks.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 77, 19 pp.

Accepted: 17 September 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K05: Renewal theory 60C05: Combinatorial probability

first passage time number of blocks regenerative composition renewal theory weak convergence

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Gnedin, Alexander; Iksanov, Alexander. Regenerative compositions in the case of slow variation: A renewal theory approach. Electron. J. Probab. 17 (2012), paper no. 77, 19 pp. doi:10.1214/EJP.v17-2002. https://projecteuclid.org/euclid.ejp/1465062399

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