The branching process in random environment 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 improve existing central limit theorem as well as we obtain Edgeworth expansions and the renewal theorem for the sequence . The strategy is to compare with partial sums of i.i.d. random variables in order to obtain precise estimates.
We thank two anonymous referees for careful reading and numerous helpful suggestions. We are also grateful to Piotr Dyszewski for useful comments and discussions. Ewa Damek incorporated some ideas communicated her by Konrad Kolesko during the work on . The research was partially supported by the National Science Center, Poland (grant number 2019/33/ B/ST1/00207).
"Limit theorems for supercritical branching processes in random environment." Bernoulli 28 (3) 1602 - 1624, August 2022. https://doi.org/10.3150/21-BEJ1349