Convergence in L^p and its exponential rate for a branching process in a random environment

We consider a supercritical branching process (Z_n) in a random environment ξ. Let W be the limit of the normalized population size Wⁿ = Z_n=E[Z_n]ξ]. We first show a necessary and sufficient condition for the quenched L^p (p > 1) convergence of (Wn), 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 ρ > 1 such that ρn(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.