Convergence rates for a branching process in a random environment

Let $(Z_n)$ be a supercritical branching process in a random environment $\xi$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| \xi]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \log W_1^{\alpha+1} < \infty$ for some $\alpha >0$, we have $W-W_n = o (n^{-\alpha})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there are norming constants $a_n(\xi)$ (that we calculate explicitly) such that $a_n(\xi) (W-W_n)$ converges in law to a non-degenerate distribution. (3) For a branching process in a finite state random environment, if $W_1$ has a finite exponential moment, then so does $W$, and the decay rate of $P(|W-W_n| > \epsilon)$ is supergeometric.