Deviations for random sums indexed by the generations of a branching process

Applying the results about harmonic moments of classical Galton-Watson process, we obtain the deviations for random sums indexed by the generations of a branching process. Our results show that the decay rates of large deviations and moderate deviations depend heavily on the degree of the heavy tail and the asymptotic distributions depend heavily on the normalizing constants. If the underlying Galton-Watson process belongs to the Schr?der case, both large deviation and moderate deviation probabilities show three decay rates, where the critical case depends heavily on the Schr?der index. Else if the Galton-Watson process belongs to the B?ttcher case, there are only two decay rate for both large deviation and moderate deviation probabilities. Simulations are also given to illustrate our results.