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2007 Record Indices and Age-Ordered Frequencies in Exchangeable Gibbs Partitions
Robert Griffiths, Dario Spano
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Electron. J. Probab. 12: 1101-1130 (2007). DOI: 10.1214/EJP.v12-434

Abstract

The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence of record indices (i.e. the least elements) of the blocks of the partition. In particular we show that, conditionally on the record indices, the distribution of the age-ordered frequencies has a left-neutral stick-breaking structure. Such a property in fact characterizes the Gibbs family among all exchangeable partitions and leads to further interesting results on: (i) the conditional Mellin transform of the $k$-th oldest frequency given the $k$-th record index, and (ii) the conditional distribution of the first $k$ normalized frequencies, given their sum and the $k$-th record index; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens' partitions.

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Robert Griffiths. Dario Spano. "Record Indices and Age-Ordered Frequencies in Exchangeable Gibbs Partitions." Electron. J. Probab. 12 1101 - 1130, 2007. https://doi.org/10.1214/EJP.v12-434

Information

Accepted: 25 August 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1148.60002
MathSciNet: MR2336601
Digital Object Identifier: 10.1214/EJP.v12-434

Subjects:
Primary: 60C05
Secondary: 60G57 , 62E10

Keywords: Age-ordered frequencies , Beta-Stacy distribution , Exchangeable Gibbs Partitions , GEM distribution , Neutral distributions , Record indices

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