In this paper we study the large deviation behavior of sums of i.i.d. random
variables X_i defined on a supercritical Galton-Watson process Z. We assume the
finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of
the partial sums of the X_i and the lower deviation probabilities of Z is
clarified. Here we heavily use lower deviation probability results on Z we
recently published in [FW06]
