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

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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.

MSC

60J80
60J05
60J22
65H10

Keywords

Multitype branching process
Extinction probability
Pathwise approach
Extinction criterion

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