Deviation inequalities for a supercritical branching process in a random environment

TL;DR

This paper derives deviation inequalities for the logarithm of the ratio of population sizes in a supercritical branching process within a random environment, with applications to confidence interval construction.

Contribution

It introduces new deviation inequalities for supercritical branching processes in random environments, enabling better probabilistic bounds and statistical inference.

Findings

Derived deviation inequalities for 1n 2 Z_n

Provided applications for confidence interval construction

Enhanced understanding of fluctuations in branching processes

Abstract

Let $\left \{ Z_{n}, n\ge 0 \right \}$ be a supercritical branching process in an independent and identically distributed random environment $\xi =\left ( \xi _{n} \right )_{n\geq 0} $. In this paper, we get some deviation inequalities for $\ln \left (Z_{n+n_{0} } / Z_{n_{0} } \right ).$ And some applications are given for constructing confidence intervals.