  Let $Z_{n}$ be the number of individuals in a subcritical BPRE evolving in
the environment generated by iid probability distributions. Let $X$ be the
logarithm of the expected offspring size per individual given the environment.
Assuming that the density of $X$ has the form $$p_{X}(x)=x^{-\beta
-1}l_{0}(x)e^{-\rho x}$$ for some $\beta >2,$ a slowly varying function
$l_{0}(x)$ and $\rho \in \left( 0,1\right),$ we find the asymptotic survival
probability and prove a Yaglom type conditional limit theorem for the process.
The survival probability decreases exponentially with an additional polynomial
term related to the tail of $X$. The proof relies on a fine study of a random
walk (with negative drift and heavy tails) conditioned to stay positive until
time $n$ and to have a small positive value at time $n$, with $n$ tending to
infinity.
