Let $\{ Z_{ 1,n} , n\geq 0\}$ and $\{ Z_{ 2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that $\frac{ 1}{ n} \ln Z_{ 1,n} \rightarrow \mu_1$ and $\frac{ 1}{ m} \ln Z_{ 2,m} \rightarrow \mu_2$ in probability as $m, n \rightarrow \infty.$ In this paper, we are interested in the comparison on the two criticality parameters. To this end, we prove a non-uniform Berry-Esseen's bound and Cram\'{ e}r's moderate deviations for $\frac{ 1}{ n} \ln Z_{ 1,n} - \frac{ 1}{ m} \ln Z_{ 2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence interval for $\mu_1-\mu_2$.
Categories: math.PR math.ST stat.TH