Skip to main content Accessibility help
×
Home

Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

  • S. Hautphenne (a1), G. Latouche (a2) and G. Nguyen (a3)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Extinction Probabilities of Branching Processes with Countably Infinitely Many Types
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Extinction Probabilities of Branching Processes with Countably Infinitely Many Types
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Extinction Probabilities of Branching Processes with Countably Infinitely Many Types
      Available formats
      ×

Copyright

Corresponding author

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

References

Hide All
[1] Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Prob. 46, 463478.
[2] Gantert, N., Müller, S., Popov, S. and Vachkovskaia, M. (2010). Survival of branching random walks in random environment. J. Theoret. Prob. 23, 10021014.
[3] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
[4] Hautphenne, S. (2012). Extinction probabilities of supercritical decomposable branching processes. J. Appl. Prob. 49, 639651.
[5] 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.
[6] Latouche, G., Nguyen, G. T. and Taylor, P. G. (2011). Queues with boundary assistance: The effects of truncation. Queueing Systems 69, 175197.
[7] Mode, C. J. (1971). Multi-Type Branching Processes: Theory and Application. Elsevier, New York.
[8] Moy, S.-T. C. (1967). Ergodic properties of expectation matrices of a branching process with countably many types. J. Math. Mech. 16, 12071225.
[9] 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, 992999.
[10] Moyal, J. E. (1962). Multiplicative population chains. Proc. R. Soc. London A. 266, 518526.
[11] Müller, S. (2008). A criterion for transience of multidimensional branching random walk in random environment. Electron. J. Prob. 13, 11891202.
[12] Sagitov, S. (2013). Linear-fractional branching processes with countably many types. Stoch. Process. Appl. 123, 29402956.
[13] Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.
[14] Spataru, A. (1989). Properties of branching processes with denumerably many types. Romanian J. Pure Appl. Math. 34, 747759.
[15] Tetzlaff, G. T. (2003). Criticality in discrete time branching processes with not uniformly bounded types. Rev. Mat. Apl. 24, 2536.
[16] 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, 847851.
[17] Zucca, F. (2011). Survival, extinction and approximation of discrete-time branching random walks. J. Statist. Phys. 142, 726753.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed