Asymptotic laws for compositions derived from transformed subordinators

Abstract

A random composition of $n$ appears when the points of a random closed set $\tilde{\mathcal{R}}\subset [0,1]$ 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 $\tilde{\mathcal{R}}=\varphi (S_{\bullet })$, where $(S_t,t\ge 0)$ is a subordinator and $\varphi :[0,\infty ]\to [0,1]$ is a diffeomorphism. We derive the asymptotics of $K_n$ when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function $\varphi (x)=1-e^{-x}$, we establish a connection between the asymptotics of $K_n$ and the exponential functional of the subordinator.

Published at http://dx.doi.org/10.1214/009117905000000639 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)