Asymptotic laws for regenerative compositions: gamma subordinators and the like

2004/05/23 by A. Gnedin, Gnedin, A., J. Pitman +3 · 1 citation
Mathematics · #60C05 #60G09 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60C05 #msc:60G09

paper · pdf · doi:10.48550/arxiv.math/0405440

arxiv created 2004/05/23 · arxiv updated 2009/12/01

Abstract

For \widetilde\cal R = 1 - exp(- \cal R) a random closed set obtained by exponential transformation of the closed range \cal R of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of \widetilde\cal R. We focus on the number of parts K_n of the composition when \widetilde\cal R is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for K_n and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+.

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