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 Cramértype 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 X_n > 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.