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

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1480388671

Digital Object Identifier
doi:10.1214/16-ECP31

Mathematical Reviews number (MathSciNet)
MR3580447

Zentralblatt MATH identifier
1355.26010

Subjects
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

Keywords
Bernstein function complete Bernstein function subordinator

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [1] Alili, L., Jedidi, W., and Rivero, V.: On exponential functionals, harmonic potential measures and undershoots of subordinators. ALEA Latin Am. J. Probab. Math. Stat. 11, (2014) 711–735.
  • [2] Marchal, P.: A class of special subordinators with nested ranges. Ann. Inst. Henri Poincaré Probab. Stat. 51, (2015) 533–544.
  • [3] Schilling, R.L., Song, R., and Vondraček, Z.: Bernstein Functions. Theory and Applications (2nd edn). De Gruyter, Studies in Mathematics 37, Berlin 2012.
  • [4] Stein, E.M. and Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton Lectures in Analysis III, Princeton (NJ) 2005.