On the survival probability for a class of subcritical branching processes in random environment

Let Z(n) be the number of individuals in a subcritical Branching Process in Random Environment (BPRE) evolving in the environment generated by i.i.d. probability distributions. Let X be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of X has the form px (x) = x(-beta-1)l(0)(x)e(-rho x) for some,8 > 2, a slowly varying function 10(x) and rho is an element of (0, 1), we find the asymptotic of the survival probability P(Z(n) > 0) as n -> infinity, prove a Yaglom type conditional limit theorem for the process and describe the conditioned environment. The survival probability decreases exponentially with an additional polynomial term related to the tail of X. The proof uses in particular a fine study of a random walk (with negative drift and heavy tails) conditioned to stay positive until time n and to have a small positive value at time n, with n -> infinity.