Precise large deviation estimates for branching process in random environment

We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $mathbb{P}[Z_n > e^{rho n}]$. Moreover in the subcritical case, under the Cramer condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.