On large deviation rates for sums associated with Galton-Watson processes

Given a super-critical Galton-Watson process $\{Z_n\}$ and a positive sequence $\{\epsilon_n\}$, we study the limiting behaviors of $P(S_{Z_n}/Z_n\geq\epsilon_n)$ and $P(S_{Z_n}/m^n\geq\epsilon_n)$ with sums $S_{n}$ of i.i.d. random variables $X_i$ and $m=E[Z_1]$. We assume that we are in Schr\"oder case with $EZ_1\log Z_1<\infty$ and $X_1$ is in the domain of attraction of an $\alpha$-stable law with $0<\alpha<2$. As by-products, when $Z_1$ is sub-exponentially distributed, we further obtain the convergence rates of $\frac{Z_{n+1}}{Z_n}$ to $m$ as $n\rightarrow\infty$.