Limit theorems for supercritical branching processes in random environment

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we obtain Edgeworth expansions and the renewal theorem for the sequence $\{\log Z_n\}_{n\ge 0}$ as well as we essentially improve the central limit theorem. Our strategy is to compare $\log Z_n$ with partial sums of i.i.d. random variables in order to obtain precise estimates.