Asymptotic laws for compositions derived from transformed subordinators

Details Asymptotic laws for compositions derived from transformed subordinators Asymptotic laws for compositions derived from transformed subordinators

2004-03-25

arxiv.org/abs/math/0403438v3

Annals of Probability 2006, Vol. 34, No. 2, 468-492

Mathematics

Probability

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

Scientific paper

10.1214/009117905000000639

A random composition of $n$ appears when the points of a random closed set $\widetilde{\mathcal{R}}\subset[0,1]$ 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 $\widetilde{\mathcal{R}}=\phi(S_{\bullet})$, where $(S_t,t\geq0)$ is a subordinator and $\phi:[0,\infty]\to[0,1]$ is a diffeomorphism. We derive the asymptotics of $K_n$ when the L\'{e}vy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function $\phi(x)=1-e^{-x}$, we establish a connection between the asymptotics of $K_n$ and the exponential functional of the subordinator.

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