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

TL;DR

This paper investigates the upper bounds of tail probabilities for the scaled growth rate in supercritical branching processes within random environments, providing new probabilistic bounds using extended Hoeffding inequalities.

Contribution

It introduces upper bounds for tail functions of the growth rate in supercritical branching processes in random environments, extending Hoeffding inequalities to this context.

Findings

Derived upper bounds for tail probabilities with $x \\geq 3$

Applied extended Hoeffding inequalities to branching processes

Provided theoretical bounds for the growth rate tail functions

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.