Journal of Theoretical Probability

, Volume 25, Issue 3, pp 703–732

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

  • V. I. Afanasyev
  • C. Böinghoff
  • G. Kersting
  • V. A. Vatutin
Article

DOI: 10.1007/s10959-010-0331-6

Cite this article as:
Afanasyev, V.I., Böinghoff, C., Kersting, G. et al. J Theor Probab (2012) 25: 703. doi:10.1007/s10959-010-0331-6

Abstract

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

Keywords

Branching process Random environment Random walk Change of measure Survival probability Functional limit theorem 

Mathematics Subject Classification (2000)

60J80 60G50 60F17 

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • V. I. Afanasyev
    • 1
  • C. Böinghoff
    • 2
  • G. Kersting
    • 2
  • V. A. Vatutin
    • 1
  1. 1.Department of Discrete MathematicsSteklov Mathematical InstituteMoscowRussia
  2. 2.Fachbereich MathematikUniversität FrankfurtFrankfurt am MainGermany

Personalised recommendations