Large Deviation Rates for Branching Processes--I. Single Type Case

Let {Zn}.0 be a Galton-Watson branching process with offspring distribution {pj}.0. We assume throughout that p0=0,pj.1 for any j.1 and 1<m=.jpj<.. Let Wn=Znm.m and W=limnWn. In this paper we study the rates of convergence to zero as n.. of P(..Zn+1Zn.m..>.),P(|Wn.W|>.), P(..Zn+1Zn.m.>...W.a) for .>0 and a>0 under various moment conditions on {pj}. It is shown that the rate for the first one is geometric if p1>0 and supergeometric if p1=0, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.