Let (Zn) be a supercritical branching process in an independent and
identically distributed random environment ξ. We show the exact decay rate
of the probability P(Zn=j∣Z0=k) as n→∞, for each j≥k, assuming that P(Z1=0)=0. We also determine the
critical value for the existence of harmonic moments of the random variable
W=limn→∞E(Zn∣ξ)Zn under a simple moment
condition