Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Let (Z_n) be a superadditive bisexual branching process in an independent and identically distributed (i.i.d.) random environment ξ. We establish Cramér moderate deviations and Berry-Esseen bounds for log(Z_n+n0=Z_n0) uniformly in n₀ ≥ 0. As an application of Cramér moderate deviation, we give the confidence interval of the parameter E log r₁(ξ₀) in terms of known population sizes Z_n0, Z_n+n0, and generation n, where r₁(ξ_n) denotes the average number of mating units formed by the offspring of the same mating unit of generation n when the environment ξ is given.