Upper bound for the tail functions of the growth rate for supercritical branching processes in random environment

Abstract

Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for $\displaystyle\mathbb{P}\left[\frac{\log Z_n}{Mn}-\mu\geq x\right]$ for any $x\geq3$ are obtained, by applying an extension of the Hoeffding type inequalities.