On the maximum of a subcritical branching process in a random environment

Let {ξn} be a subcritical branching process in random environment with independent identically distributed generating functions fn(s). It is shown that if there exists a positive number æ such that E(f0′(1))æ=1 then, for x→+∞,Psupn ξn>x∼Kx−æ,where K is a positive constant.