Convergence in Lp and its exponential rate for a branching process in a random environment

We consider a supercritical branching process (Z(n)) in a random environment xi. Let W be the limit of the normalized population size W-n = Z(n)/E[Z(n)|xi]. We first show a necessary and sufficient condition for the quenched L-p (p > 1) convergence of (W-n), which completes the known result for the annealed L-p convergence. We then show that the convergence rate is exponential, and we find the maximal value of rho > 1 such that rho(n)(W - W-n) --> 0 in L-p, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment.