Uniform Cramér moderate deviations and Berry-Esseen bounds for superadditive bisexual branching processes in random environments (Q6592727)

Given a random environment $\xi=(\xi_n)_{n\in \mathbb{N}}$, $\xi_n$ with probability distribution $p(\xi_n) = \{ p_i(\xi_n); i\in \mathbb{N}\}$ on $\mathbb{N}$ with $\sum_i ip_i(\xi_n)\in (0,\infty)$, the branching process considered is defined by $Z_0 = 1$, $Z_{n+1}= L ( \sum_{i=1}^{Z_n} (f_{ni},m_{ni}))$, $n\in \mathbb{N}$, where $(f_{ni},m_{ni}), i \geq 1$, are conditionally i.i.d., given $\xi$, $f_{ni}+m_{ni}$ with distribution $p(\xi_n)$, and $L: \mathbb{N}^2\to\mathbb{N}$ superadditive with $L(x,0) = L(0,y)$ for all $x$ and $y$. Here, $f_{ni}$ and $m_{ni}$ represent the number of female, respectively, male offsprings produced in the $i$'th mating unit in the $n$'th generation. Let $X:= \log E_\xi (Z_1|Z_0 =1)$, where $E_\xi$ denotes the conditional expectation, given $\xi$. With sufficient moment assumptions, the authors obtain for $G_{l,n}:= \sigma^{-1}n^{-1/2} (\log (Z_{l+n}/Z_0)-n\mu)$, $l,n\in \mathbb{N}$, $\mu:= E X > 0$, $\sigma:= \operatorname{Var} X \in (0,\infty)$, uniform Cramér deviation results, e.g., $| \log (P(G_{n,l} \geq x)/(1-\phi(x)))| \leq C(1+x^3)n^{-1/2} \log n$ uniformly in $l \in \mathbb{N}$ for $n \geq 2$ and $0 \leq x \leq (\log n)^{1/2}$, uniform Berry-Esseen bounds, e.g., $\sup_{x\in \mathbb{R}} |P(G_{l,n} \leq x) - \phi(x)| \leq n^{-1/2}\log n$, and similar results for $- G_{l,n}$, with $\phi$ denoting the normal distribution function. The Cramér moderate deviation results are applied to construct confidence intervals for $\mu$.