Let {Zn, n ³ 1} be a
single type supercritical Galton-Watson process with mean
EZ1 º m, initiated by a single ancestor.
This paper studies the large
deviation behavior of the sequence {Rn º Zn+1/Zn:
n ³ 1} and establishes a "phase
transition" in rates depending on whether r, the maximal number
of moments possessed by the offspring distribution, is less than, equal
to, or greater than the Schroder constant a. This is done via a careful analysis of the
harmonic moments of Zn.