Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment

Abstract: Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\xi$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$ assuming that $\mathbb{P} (Z_1 = 0) =0$. We also determine the critical value for the existence of harmonic moments of the random variable $W=\lim_{n\to\infty}\frac{Z_n}{\mathbb E (Z_n|\xi)}$ under a simple moment condition.