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

TL;DR

This paper investigates the asymptotic behavior of large deviation probabilities for sums associated with super-critical Galton-Watson processes, especially under stable law attraction and Schröder case conditions.

Contribution

It provides new insights into large deviation rates for sums over Galton-Watson processes in the Schröder case with stable law domain attraction, including convergence rates for the process ratio.

Findings

Derived asymptotic behaviors of large deviation probabilities

Established convergence rates of $Z_{n+1}/Z_n$ to $m$ under sub-exponential distributions

Analyzed sums of i.i.d. variables in the domain of attraction of stable laws

Abstract

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 $n\rightarrow\infty$.