Gaweł and Kimmel (1996) introduced and studied iterated Galton–Watson processes, (Xn)n≥0, possibly with thinning, as models of the number of repeats of DNA triplets during some genetic disorders. Our main results are the following. If the process indeed involves some thinning then extinction, {Xn→0}, and explosion, {Xn→∞}, can have positive probability simultaneously. If the underlying (simple) Galton–Watson process is nondecreasing with mean m then, conditionally on explosion, the ratios (log Xn+1)/Xn converge to logm almost surely. This simplifies the arguments of Gaweł and Kimmel, and confirms and extends a conjecture of Pakes (2003).
