Asymptotic laws for compositions derived from transformed subordinators

Abstract

A random composition of n appears when the points of a random closed set R̃ ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K n of this composition and other related functionals under the assumption that, R̃ = Φ(S •), where (St, t ≥ 0) is a subordinator and Φ: [0, ∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of K n when the Levy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function Φ (x) = 1 - e -x, we establish a connection between the asymptotics of K n and the exponential functional of the subordinator. © Institute of Mathematical Statistics, 2006.