Asymptotics of iterated branching processes

We study iterated Galton-Watson processes, introduced by Gawel and Kimmel as models of the number of repeats of DNA triplets during some genetic disorders. When the process (Xn)n involves some thinning, the extinction Xn → 0 and the explosion Xn → ∞ can have positive probability simultaneously. When the underlying (simple) Galton-Watson process is nondecreasing with mean m, conditional on the explosion, log Xn+1 ∼ Xn · log m almost surely. This simplifies arguments of Gawel and Kimmel, and confirms and extends a conjecture of Pakes