Comparison on the criticality parameters for two supercritical branching processes in random environments

Abstract

Let $\{Z_{1,n},n\ge 0\}$ and $\{Z_{2,n},n\ge 0\}$ be two supercritical branching processes in different random environments, with criticality parameters ${\mu }_1$ and ${\mu }_2$ respectively. It is known that $\frac{1}{n}\ln Z_{1,n}\to {\mu }_1$ and $\frac{1}{m}\ln Z_{2,m}\to {\mu }_2$ in probability as $m,n\to \infty .$ In this paper, we are interested in the comparison on the two criticality parameters. To this end, we prove a non-uniform Berry-Esseen's bound and Cramér's moderate deviations for $\frac{1}{n}\ln Z_{1,n}-\frac{1}{m}\ln Z_{2,m}$ as $m,n\to \infty .$ An application is also given for constructing confidence interval for ${\mu }_1-{\mu }_2$.

38 pages