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

Abstract

Suppose that $(Z_n{)}_{n\ge 0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n{)}_{n\ge 0}$ is studied. The upper bounds for $\mathbb{P}\left[\frac{\log Z_n}{Mn}-\mu \ge x\right]$ for any $x\ge 3$ are obtained, by applying an extension of the Hoeffding type inequalities.

Some typos are spotted out and some errors are found in the proof of the results in the first version. This is the updated version