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 $\xi$. Let $W$ be the limit of the normalized population size $W_n=Z_n\mathrm{/}E[Z_n\mathrm{\mid }\xi ]$. 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 ${\rho }^n(W-W_n)\to 0$ in $L^p$, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment.