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We consider the extinction events of Galton-Watson processes with countablyinfinitely many types. In particular, we construct truncated and augmentedGalton-Watson processes with finite but increasing sets of types. A pathwiseapproach is then used to show that, under some sufficient conditions, thecorresponding sequence of extinction probability vectors converges to theglobal extinction probability vector of the Galton-Watson processes withcountably infinitely many types. This gives rise to a number of iterativemethods for the computation of the global extinction probability vector.

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