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Asymptotic laws for compositions derived from transformed subordinators. (English) Zbl 1142.60327

Summary: 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 \((S_t\), \(t\geq 0)\) is a subordinator and \(\phi: [0,\infty]\to[0,1]\) is a diffeomorphism. We derive the asymptotics of Kn when the Lévy 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.

MSC:

60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
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