Self-Normalized Cramér-Type Moderate Deviations for the Harris Estimator of a Supercritical Branching Process

Let (Xn)n≥0 be a supercritical Galton–Watson process with offspring mean m>1, positive variance, and E|X1−m|2+ρ<∞ for some 0<ρ≤1. We establish self-normalized Cramér-type moderate deviations for the Studentized Harris estimator. It is shown that there exists a constant c>0 so that the logarithmic relative error in normal approximation is of order O(e−cn), uniformly for 0≤x≤cn. The self-normalized Cramér-type moderate deviations hold with condition p0=0, including the Böttcher case p0=p1=0. When p0>0, similar results hold conditional on non-extinction or on Xn>0. Consequently, we obtain moderate deviation principles, Berry–Esseen bounds, and asymptotic confidence intervals. Numerical experiments for bounded and unbounded offspring laws with p0=0, p1>0 examine tail approximation and interval coverage.