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

Abstract

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\"{o}der case, both large deviation  and moderate deviation probabilities show three decay rates, where the  critical case depends heavily on the Schr\"{o}der index. Else if the Galton--Watson process belongs to the B\"{o}ttcher case, there are only two decay rates for both  large deviation  and moderate deviation probabilities. Simulations   are also given to illustrate our results.