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

Let (Zn)(n >= 0) be a supercritical branching process in an independent and identically distributed random environment xi = (xi(n))(n >= 0). We study the asymptotic behavior of the harmonic moments E[Z(n)(-r)vertical bar Z(0) = k] of order r > 0 as n -> infinity, when the process starts with k initial individuals. We exhibit a phase transition with the critical value r(k) > 0 determined by the equation Ep(1)(k) (xi(-rk)(0), where m(0) = Sigma(infinity)(j=0) jpj (xi(0)) (p(j) (xi(0)))(j >= 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 ->infinity) Zn/E(Z(n)vertical bar 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).