Quenched weighted moments of a supercritical branching process in a random environment

We consider a supercritical branching process (Zn) in an independent and identically distributed random environment = (n). Let W be the limit of the natural martingale Wn = Zn/E Zn, n 0, where E denotes the conditional expectation given the environment . We find a necessary and sufficient condition for the existence of quenched weighted moments of W of the form E W l(W ), where > 1 and l is a positive function slowly varying at . The same conclusion is also proved for the maximum of the martingale W * = sup n1 Wn instead of the limit variable W . In the proof we first show an extended version of Doob's inequality about weighted moments for nonnegative submartingales, which is of independent interest.