Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment (Q6650754)

Suppose to be given a discrete-time, one-type branching process $(Z_n)_{n\geq0}$ with random environment $\xi=(\xi_n)_{n\geq0}$. The $\xi_n, n=0,1,2,\dots$ are i.i.d. and each realization $\xi_n$ corresponds to a probability distribution $p(\xi_n)=\{p_i(\xi_n); i=0,1,2,\dots\}$ on the non-negative intergers, with $\sum_i i p_i (\xi_n) \in (0,\infty)$. Let $m_n:= \sum_{0\leq k<\infty} kp_k(\xi_n)$ and $I_n:= \sum_{0\leq k\leq n-1} (Z_{k+1}/Z_k)$. \N\NThe authors investigate the martingale $U_n:= \sum_{0\leq k\leq n-1}(Z_{k+1}/Z_k -m_k), n\geq1$, deriving, in particular, an exponential convergence rate for it in $L^1$, whenever $\mathbb{E} |Z_1-m_0|^p < \infty$ for some $p > 1$. Assuming $\nu:=\mathbb{E}m_0 < \infty$, $\sigma^2:=\operatorname{Var} m_0 \in (0,\infty)$, $\mathbb{E}Z_1^3< \infty$, and $m_0$ to be non-lattice, an explicit expression for the limit of $n^{1/2}[\mathbb{P}((I_n - n\nu)/(n^{1/2}\sigma) \leq x) - \Phi(x)]$, as $n\to\infty$, is given (where $\Phi$ denotes the normal distribution function).