Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment $(Z_n\colon n\geq0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of $Z$, which means the asymptotic behavior of the probability $\{1\leq Z_n\leq\exp(n\theta)\}$ as $n\to\infty$. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where $\mathbb P(Z_1=0\mid Z_0=1)>0$ and also much weaker moment assumptions.