Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-30T21:47:48.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour near extinction of continuous-state branching processes

Part of: Stochastic processes

Published online by Cambridge University Press:  21 June 2016

Gabriel Berzunza  and
Juan Carlos Pardo
Gabriel Berzunza*
Affiliation:
Universität Zürich
Juan Carlos Pardo*
Affiliation:
Centro de Investigación en Matemáticas
*
* Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: gabriel.berzunza@math.uzh.ch
** Postal address: Centro de Investigación en Matemáticas, A.C. Calle Jalisco s/n, 36240 Guanajuato, México. Email address: jcpardo@cimat.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

Keywords

Continuous-state branching processLamperti transformLévy processconditioning to stay positiverate of growth

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G51: Processes with independent increments; L'evy processes

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 381 - 391
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Bertoin, J. (1996).Lévy Processes.Cambridge University Press.Google Scholar
[2]Bingham, N. H. (1976).Continuous branching processes and spectral positivity.Stoch. Process. Appl. 4, 217–242.Google Scholar
[3]Duquesne, T. (2012).The exact packing measure of Lévy trees.Stoch. Process. Appl. 122, 968–1002.CrossRefGoogle Scholar
[4]Grey, D. R. (1974).Asymptotic behaviour of continuous-time, continuous state-space branching processes.J. Appl. Prob. 11, 669–677.Google Scholar
[5]Grimvall, A. (1974).On the convergence of sequences of branching processes.Ann. Prob. 2, 1027–1045.CrossRefGoogle Scholar
[6]Jiřina, M. (1958).Stochastic branching processes with continuous state space.Czech. Math. J. 8, 292–313.Google Scholar
[7]Kyprianou, A. E. and Pardo, J. C. (2008).Continuous-state branching processes and self-similarity.J. Appl. Prob. 45, 1140–1160.Google Scholar
[8]Lamperti, J. (1967).Continuous state branching processes.Bull. Amer. Math. Soc. 73, 382–386.Google Scholar
[9]Pardo, J. C. (2008).On the rate of growth of Lévy processes with no positive jumps conditioned to stay positive.Electron. Commun. Prob. 13, 494–506.Google Scholar
[10]Silverstein, M. L. (1968).A new approach to local times.J. Math. Mech. 17, 1023–1054.Google Scholar