Let \(\{Z_n\}\) be a critical Galton–Watson process and \(S_{Z_n}:=\sum _{k=1}^{Z_n}X_k\) be sums of i.i.d. random variables \(\{X_k\}\) . In this paper, we study the asymptotic behavior of \(S_{Z_n}/Z_n\) conditioned on \(\{Z_n>0\}\) . A “phase transition” in rates is identified for the large deviation, depending on whether \(\beta \) , the negative index of the regularly varying right tail of \(X_1\) , is less than, equal to or greater than 2. The self-normalized large deviation is also established without any moment assumption on \(X_1\) . Moreover, Berry–Esseen bounds for the Lotka–Nagaev estimator, namely \(Z_{n+1}/Z_n\) , are derived by Stein’s method. As a by-product, new rates of convergence for Yaglom’s theorem are obtained.
