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

Let ${\left(X_n\right)}_{n\ge 0}$ be a supercritical Galton–Watson process with offspring mean $m>1$, positive variance, and $\mathbb{E}|X_1{-m|}^{2+\rho }<\infty$ for some $0<\rho \le 1$. 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 $O\left(e^{-cn}\right)$, uniformly for $0\le x\le c\sqrt{n}$. The self-normalized Cramér-type moderate deviations hold with condition $p_0=0$, including the Böttcher case $p_0=p_1=0$. When $p_0>0$, 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 $p_0=0$, $p_1>0$ examine tail approximation and interval coverage.