2016/08/29 by Ion Grama, Quansheng Liu, Grama, Ion +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1608.08075
arxiv created 2016/08/29 · arxiv updated 2016/08/30
Let (Zn) be a supercritical branching process in an independent and identically distributed random environment ξ. We study the asymptotic of the harmonic moments 𝔼[Zn-r | Z0=k ] of order r>0 as n → ∞. We exhibit a phase transition with the critical value rk>0 determined by the equation \mathbb E p1k = \mathbb E m0-rk, where m0=∑k=0^∞ k pk with pk=\mathbb P(Z1=k | ξ), assuming that p0=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=limn→∞ Zn / \mathbb E (Zn|ξ). The aforementioned phase transition is linked to that for the rate function of the lower large deviation for Zn. As an application, we obtain a lower large deviation result for Zn 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-Wn, and find an equivalence for the large deviation probabilities of the ratio Zn+1 / Zn.