Electronic Communications in Probability

Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal

Chang-Song Deng and René L. Schilling

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Let $\alpha :[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal [2] that the function \[ \phi ^{(\alpha )}(\lambda ):=\exp \left [ \int _0^1\frac{\lambda -1} {1+(\lambda -1)x}\,\alpha (x)\,\mathrm{d} x \right ],\quad \lambda >0 \] is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(\alpha )}$ such that $\mathcal{R} ^{(\alpha )} \stackrel{\text {law}} {=} \overline{\{S^{(\alpha )}_t:t\geq 0\}} $ ($S^{(\alpha )}$ is the subordinator with Laplace exponent $\phi ^{(\alpha )}$) and $\mathcal R^{(\alpha )}\subset \mathcal R^{(\beta )}$ whenever $\alpha \leq \beta $. We give two simple proofs showing that $\phi ^{(\alpha )}$ is a complete Bernstein function and extend Marchal’s construction to all complete Bernstein functions.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 78, 5 pp.

Received: 15 June 2016
Accepted: 17 November 2016
First available in Project Euclid: 29 November 2016

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Primary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 30H15: Nevanlinna class and Smirnov class 60G51: Processes with independent increments; Lévy processes

Bernstein function complete Bernstein function subordinator

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Deng, Chang-Song; Schilling, René L. Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal. Electron. Commun. Probab. 21 (2016), paper no. 78, 5 pp. doi:10.1214/16-ECP31. https://projecteuclid.org/euclid.ecp/1480388671

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