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

Given a super-critical Galton-Watson process $\{Z_n\}$ and a positive sequence $\{{ϵ}_n\}$, we study the limiting behaviors of $P(S_{Z_n}\mathrm{/}{Z}_n\ge {ϵ}_n)$ and $P(S_{Z_n}\mathrm{/}{m}^n\ge {ϵ}_n)$ 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 $E{Z}_1\log Z_1<\mathrm{\infty }$ 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 $n\to \mathrm{\infty }$.