This paper considers two supercritical branching processes with immigration in different random environments, denoted by $\{Z_{1,n}\}$ and $\{Z_{2,m}\}$, with criticality parameters mu 1 and mu 2, respectively. Under certain conditions, it is known that $\frac{1}{n} \log Z_{1,n} \to \mu_1$ and $\frac{1}{m} \log Z_{2,m} \to \mu_2$ converge in probability as $m, n \to \infty$. We present basic properties about a central limit theorem, a non-uniform Berry-Esseen's bound, and Cram & eacute;r's moderate deviations for $\frac{1}{n} \log Z_{1,n} - \frac{1}{m} \log Z_{2,m}$ as $m, n \to \infty$. To th...
