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A pathwise approach to the extinction of branching processes with countably many types

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  • Braunsteins, Peter
  • Decrouez, Geoffrey
  • Hautphenne, Sophie

Abstract

We consider the extinction events of Galton–Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton–Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton–Watson process with countably infinitely many types. Besides giving rise to a family of new iterative methods for computing the global extinction probability vector, our approach paves the way to new global extinction criteria for branching processes with countably infinitely many types.

Suggested Citation

  • Braunsteins, Peter & Decrouez, Geoffrey & Hautphenne, Sophie, 2019. "A pathwise approach to the extinction of branching processes with countably many types," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 713-739.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:3:p:713-739
    DOI: 10.1016/j.spa.2018.03.013
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    References listed on IDEAS

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    1. Sagitov, Serik, 2013. "Linear-fractional branching processes with countably many types," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2940-2956.
    2. Gibson, Diana & Seneta, E., 1987. "Monotone infinite stochastic matrices and their augmented truncations," Stochastic Processes and their Applications, Elsevier, vol. 24(2), pages 287-292, May.
    3. Jagers, Peter, 1989. "General branching processes as Markov fields," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 183-212, August.
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