Let $\left\{Z_n\right\}$ be a supercritical branching process in an independent and identically distributed random environment. As is well known, the behavior of $Z_n$ depends primarily on that of the associated random walk $S_n$ constructed by the logarithms of the quenched expectation of population sizes. By this observation, the Berry–Esséen bound for $log{Z}_n$ has been established by Grama et al. (2017). To refine that, we figure out the exact convergence rate in the central limit theorem for $log{Z}_n$ under the annealed law, with less restrictive moment conditions. In particular, there is one factor in the rate function concerning on $log{Z}_n$ that does not appear in that for $S_n$. Hence the result indicates the essential difference between $log{Z}_n$ and $S_n$.

让 $\left\{Z_n\right\}$是一个独立同分布的随机环境中的超临界分支过程。众所周知，行为$Z_n$ 主要取决于相关的随机游走 ${秒}_n$由人口规模的淬灭期望的对数构建。根据这一观察，Berry-Esséen 前往$日志Z_n$Grama 等人已经建立了。(2017)。为了完善这一点，我们计算出中心极限定理中的精确收敛速度$日志Z_n$在退火定律下，力矩条件限制较少。特别地，率函数中有一个因素涉及$日志Z_n$ 没有出现在那个中 ${秒}_n$. 因此，结果表明两者之间的本质区别$日志Z_n$ 和 ${秒}_n$.