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

Let {Z_1,n , n≥ 0} and {Z_2,n, n≥ 0} be two supercritical branching processes in different random environments, with criticality parameters μ_1 and μ_2 respectively. It is known that 1/nln Z_1,n→μ_1 and 1/mln Z_2,m→μ_2 in probability as m, n →∞. 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 Cramér's moderate deviations for 1/nln Z_1,n - 1/mln Z_2,m as m, n →∞. An application is also given for constructing confidence interval for μ_1-μ_2.

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