Deviation inequalities for a supercritical branching process in a random environment

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.