Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-29T06:12:46.048Z Has data issue: false hasContentIssue false
  • English
  • Français

Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

Part of: Nonlinear algebraic or transcendental equations Markov processes

Published online by Cambridge University Press:  04 January 2016

S. Hautphenne ,
G. Latouche  and
G. Nguyen
S. Hautphenne*
Affiliation:
The University of Melbourne
G. Latouche*
Affiliation:
Université libre de Bruxelles
G. Nguyen*
Affiliation:
The University of Adelaide
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia. Email address: sophiemh@unimelb.edu.au
∗∗ Postal address: Département d'Informatique, Université libre de Bruxelles, 1050 Brussels, Belgium. Email address: latouche@ulb.ac.be
∗∗∗ Postal address: School of Mathematical Sciences, The University of Adelaide, South Australia 5005, Australia. Email address: giang.nguyen@adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

Keywords

Multitype branching processextinction probabilityextinction criteriaiterative method

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains 65H10: Systems of equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1068 - 1082
Copyright
© Applied Probability Trust 

References

Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Prob. 46, 463–478.CrossRefGoogle Scholar
Gantert, N., Müller, S., Popov, S. and Vachkovskaia, M. (2010). Survival of branching random walks in random environment. J. Theoret. Prob. 23, 1002–1014.CrossRefGoogle Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Hautphenne, S. (2012). Extinction probabilities of supercritical decomposable branching processes. J. Appl. Prob. 49, 639–651.CrossRefGoogle Scholar
Korn, G. A. and Korn, T. M. (1961). Mathematical Handbook for Scientists and Engineers: Definition, Theorems, and Formulas for Reference and Review. McGraw-Hill, New York.Google Scholar
Latouche, G., Nguyen, G. T. and Taylor, P. G. (2011). Queues with boundary assistance: The effects of truncation. Queueing Systems 69, 175–197.CrossRefGoogle Scholar
Mode, C. J. (1971). Multi-Type Branching Processes: Theory and Application. Elsevier, New York.Google Scholar
Moy, S.-T. C. (1967). Ergodic properties of expectation matrices of a branching process with countably many types. J. Math. Mech. 16, 1207–1225.Google Scholar
Moy, S.-T. C. (1967). Extensions of a limit theorem of Everett, Ulam and Harris on multitype branching processes to a branching process with countably many types. Ann. Math. Statist. 38, 992–999.CrossRefGoogle Scholar
Moyal, J. E. (1962). Multiplicative population chains. Proc. R. Soc. London A. 266, 518–526.Google Scholar
Müller, S. (2008). A criterion for transience of multidimensional branching random walk in random environment. Electron. J. Prob. 13, 1189–1202.CrossRefGoogle Scholar
Sagitov, S. (2013). Linear-fractional branching processes with countably many types. Stoch. Process. Appl. 123, 2940–2956.CrossRefGoogle Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Spataru, A. (1989). Properties of branching processes with denumerably many types. Romanian J. Pure Appl. Math. 34, 747–759.Google Scholar
Tetzlaff, G. T. (2003). Criticality in discrete time branching processes with not uniformly bounded types. Rev. Mat. Apl. 24, 25–36.Google Scholar
van Doorn, E. A., van Foreest, N. D. and Zeifman, A. I. (2009). Representations for the extreme zeros of orthogonal polynomials. J. Comput. Appl. Math. 233, 847–851.CrossRefGoogle Scholar
Zucca, F. (2011). Survival, extinction and approximation of discrete-time branching random walks. J. Statist. Phys. 142, 726–753.CrossRefGoogle Scholar