Electronic Communications in Probability

Biggins’ martingale convergence for branching Lévy processes

Jean Bertoin and Bastien Mallein

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Abstract

A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(\sigma ^2,a,\Lambda )$, where the branching Lévy measure $\Lambda $ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins’ theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma ^2,a,\Lambda )$ for additive martingales to have a non-degenerate limit.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 83, 12 pp.

Dates
Received: 13 December 2017
Accepted: 17 October 2018
First available in Project Euclid: 25 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1540433049

Digital Object Identifier
doi:10.1214/18-ECP185

Zentralblatt MATH identifier
1402.60051

Subjects
Primary: 60G44: Martingales with continuous parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
branching Lévy process additive martingale uniform integrability spinal decomposition

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bertoin, Jean; Mallein, Bastien. Biggins’ martingale convergence for branching Lévy processes. Electron. Commun. Probab. 23 (2018), paper no. 83, 12 pp. doi:10.1214/18-ECP185. https://projecteuclid.org/euclid.ecp/1540433049


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References

  • [1] Gerold Alsmeyer and Alexander Iksanov, A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks, Electron. J. Probab. 14 (2009), no. 10, 289–312.
  • [2] J. Berestycki, S. C. Harris, and A. E. Kyprianou, Traveling waves and homogeneous fragmentation, Ann. Appl. Probab. 21 (2011), no. 5, 1749–1794.
  • [3] Jean Bertoin, Compensated fragmentation processes and limits of dilated fragmentations, Ann. Probab. 44 (2016), no. 2, 1254–1284.
  • [4] Jean Bertoin, Markovian growth-fragmentation processes, Bernoulli 23 (2017), no. 2, 1082–1101.
  • [5] Jean Bertoin and Bastien Mallein, Infinitely ramified point measures and branching Lévy processes, Ann. Probab. (2018), To appear.
  • [6] J. D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probability 14 (1977), no. 1, 25–37.
  • [7] J. D. Biggins, Uniform convergence of martingales in the branching random walk, Ann. Probab. 20 (1992), no. 1, 137–151.
  • [8] Benjamin Dadoun, Asymptotics of self-similar growth-fragmentation processes, Electron. J. Probab. 22 (2017), Paper No. 27, 30.
  • [9] Richard Durrett, Probability: theory and examples, second ed., Duxbury Press, Belmont, CA, 1996.
  • [10] Robert Hardy and Simon C. Harris, A spine approach to branching diffusions with applications to $\mathcal{L} ^p$-convergence of martingales, Séminaire de Probabilités XLII, Lecture Notes in Math., vol. 1979, Springer, Berlin, 2009, pp. 281–330.
  • [11] A. E. Kyprianou, A note on branching Lévy processes, Stochastic Process. Appl. 82 (1999), no. 1, 1–14.
  • [12] Günter Last and Mathew Penrose, Lectures on the Poisson process, Institute of Mathematical Statistics Textbooks, vol. 7, Cambridge University Press, Cambridge, 2018.
  • [13] Quansheng Liu, Fixed points of a generalized smoothing transformation and applications to the branching random walk, Adv. in Appl. Probab. 30 (1998), no. 1, 85–112.
  • [14] Russell Lyons, A simple path to Biggins’ martingale convergence for branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221.
  • [15] Pascal Maillard, Speed and fluctuations of $N$-particle branching Brownian motion with spatial selection, Probab. Theory Related Fields 166 (2016), no. 3–4, 1061–1173.
  • [16] Quan Shi and Alex Watson, Probability tilting of compensated fragmentations, arXiv:1707.00732, jul 2017.
  • [17] Kōhei Uchiyama, Spatial growth of a branching process of particles living in ${\bf R}^{d}$, Ann. Probab. 10 (1982), no. 4, 896–918.