  Branching Processes in Random Environment (BPREs) $(Z\_n:n\geq0)$ 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
$n\rightarrow \infty$. We provide an expression of the rate of decrease of this
probability, under some moment assumptions, which yields the rate function.
This result generalizes the lower large deviation theorem of Bansaye and
Berestycki (2009) by considering processes where $\P(Z\_1=0 \vert
Z\_0=1)\textgreater{}0$ and also much weaker moment assumptions.
