Harmonic moments and large deviations for a supercritical branching process in a random environment

Abstract

Let $(Z_n)_{n\geq 0}$ be a supercritical branching process in an independent and identically distributed random environment $\xi =(\xi _n)_{n\geq 0}$. We study the asymptotic behavior of the harmonic moments $\mathbb{E} \left [Z_n^{-r} | Z_0=k \right ]$ of order $r>0$ as $n \to \infty $, when the process starts with $k$ initial individuals. We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\mathbb E p_1^k(\xi _0) = \mathbb E m_0^{-r_k},$ where $m_0=\sum _{j=0}^\infty j p_j (\xi _0)$, $(p_j(\xi _0))_{j\geq 0}$ being the offspring distribution given the environnement $\xi _0$. Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence of the harmonic moments of $W=\lim _{n\to \infty } Z_n / \mathbb E (Z_n|\xi ).$ The aforementioned phase transition is linked to that for the rate function of the lower large deviation for $Z_n$. As an application, we obtain a lower large deviation result for $Z_n$ under weaker conditions than in previous works and give a new expression of the rate function. We also improve an earlier result about the convergence rate in the central limit theorem for $W-W_n,$ and find an equivalence for the large deviation probabilities of the ratio $Z_{n+1} / Z_n$.