Self-Normalized Cramér Type Moderate Deviations for Pooled Estimation in Branching Processes in a Random Environment

We study the estimation of the offspring mean of a supercritical branching process in a random environment when multiple conditionally independent populations evolve in a common environment. Extending the single-population self-normalized Cramér moderate deviation theory to this multi-population setting, we introduce a pooled Lotka–Nagaev estimator and construct its associated martingale difference sequence. The core of the analysis is an exact decomposition of the pooled conditional variance, which, owing to the shared environment and conditional independence of the populations, separates the environmental and demographic sources of variability. This decomposition reveals a structural dichotomy: the environmental variance is undiluted by pooling, while the demographic variance is attenuated at a rate proportional to the inverse square of the number of populations. Verifying the two conditions of the martingale moderate-deviation theorem yields self-normalized Cramér moderate deviations for the pooled Student t-statistic, together with a Berry–Esseen bound, a moderate deviation principle, and confidence intervals for the offspring mean. The resulting pooling efficiency gain, in which the demographic variance decays inversely with the number of populations while the environmental variance forms an irreducible floor, has no analogue in any single-population framework and is confirmed by Monte Carlo simulation.