Regenerative composition structures

Regenerative composition structures

Alexander Gnedin and Jim Pitman

Source: Ann. Probab. Volume 33, Number 2 (2005), 445-479.

Abstract

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterization results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0,1] generated by excursions of a standard Bessel bridge of dimension 2−2α for some α∈[0,1].

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Primary Subjects: 60G09, 60C05

Keywords: Exchangeability; composition structure; regenerative set; sampling formula; subordinator

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868588
Digital Object Identifier: doi:10.1214/009117904000000801
Mathematical Reviews number (MathSciNet): MR2122798
Zentralblatt MATH identifier: 02164470

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