Journal of Applied Probability

Extinction probabilities of supercritical decomposable branching processes

Sophie Hautphenne

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Article information

J. Appl. Probab., Volume 49, Number 3 (2012), 639-651.

First available in Project Euclid: 6 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Multitype branching process decomposable branching process extinction criteria extinction probability


Hautphenne, Sophie. Extinction probabilities of supercritical decomposable branching processes. J. Appl. Probab. 49 (2012), no. 3, 639--651. doi:10.1239/jap/1346955323.

Export citation


  • Bean, N. G., Kontoleon, N. and Taylor, P. G. (2008). Markovian trees: properties and algorithms. Ann. Operat. Res. 160, 31–50.
  • Foster, J. and Ney, P. (1976). Decomposable critical multi-type branching processes. Sankhyā A 38, 28–37.
  • Foster, J. and Ney, P. (1978). Limit laws for decomposable critical branching processes. Z. Wahrscheinlichkeitsth. 46, 13–43.
  • Gantmacher, F. R. (1974). The Theory of Matrices. Chelsa Publishing, New York.
  • Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • Hautphenne, S., Latouche, G. and Remiche, M.-A. (2008). Newton's iteration for the extinction probability of a Markovian binary tree. Linear Algebra Appl. 428, 2791–2804.
  • Hautphenne, S., Latouche, G. and Remiche, M.-A. (2011). Algorithmic approach to the extinction probability of branching processes. Methodology Comput. Appl. Prob. 13, 171–192.
  • Kesten, H. and Stigum, B. P. (1966). A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37, 1211–1223.
  • Kesten, H. and Stigum, B. P. (1967). Limit theorems for decomposable multi-dimensional Galton–Watson processes. J. Math. Anal. Appl. 17, 309–338.
  • Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications. Elsevier, New York.
  • Olofsson, P. (2000). A branching process model of telomere shortening. Commun. Statist. Stoch. Models 16, 167–177.
  • Scalia-Tomba, G.-P. (1986). The asymptotic final size distribution of reducible multitype Reed–Frost processes. J. Math. Biol. 23, 381–392.
  • Sewastjanow, B. A. (1975). Verzweigungsprozesse. R. Oldenbourg, Munich.
  • Sugitani, S. (1979). On the limit distribution of decomposable Galton–Watson processes. Proc. Japan Acad. 55, 334–336.