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A special family of Galton-Watson processes with explosions
The linear-fractional Galton-Watson processes is a well known case when many
characteristics of a branching process can be computed explicitly. In this
paper we extend the two-parameter linear-fractional family to a much richer
four-parameter family of reproduction laws. The corresponding Galton-Watson
processes also allow for explicit calculations, now with possibility for
infinite mean, or even infinite number of offspring. We study the properties of
this special family of branching processes, and show, in particular, that in
some explosive cases the time to explosion can be approximated by the Gumbel
distribution