Comparison on the criticality parameters for two supercritical branching processes with immigration in random environments

This paper considers two supercritical branching processes with immigration in different random environments, denoted by 1,n\ and 2,m\ , with criticality parameters µ 1 and µ 2 , respectively. Under certain conditions, it is known that 1/n log Z1,n → μ₁ and 1/m log Z2,m → μ₂ converge in probability as m, n → ∞ . We present basic properties about a central limit theorem, a non-uniform Berry–Esseen’s bound, and Cramér’s moderate deviations for 1/n log Z1,n - 1/m log Z2,m as m, n → ∞ . To this end, applications to construction of confidence intervals and simulations are also given.