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

TL;DR

This paper compares the criticality parameters of two supercritical branching processes in random environments, providing bounds and deviations to statistically distinguish their growth rates.

Contribution

It introduces non-uniform Berry-Esseen bounds and Cramér's moderate deviations for the difference of the logarithmic growth rates of two processes, enabling confidence interval construction.

Findings

Established bounds for the difference in criticality parameters.

Derived moderate deviation results for the difference.

Provided a method for confidence interval estimation.

Abstract

Let $\{Z_{1,n} , n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that $\frac{1}{n} \ln Z_{1,n} \rightarrow \mu_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow \mu_2$ in probability as $m, n \rightarrow \infty.$ 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 Cram\'{e}r's moderate deviations for $\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence interval for $\mu_1-\mu_2$.