\label{linear}$$where $p_{n}+q_{n}=1,\ p_{n}q_{n}>0.$

It turns out that, many (but not all) of the forthcoming results in this paper can be proved for the branching processes in random environment respecting Assumptions $A1-A2$ only. However, the main advantage we gain by imposing Assumption $A3$ is that if we let$$\begin{aligned}
f_{k,n}\left( s\right) &:= &f_{k}(f_{k+1}(...(f_{n-1}\left( s\right)
)...)),\;0\leq k\leq n-1,\;f_{n,n}\left( s\right) :=s,  \notag \\
f_{n,m}\left( s\right) &:= &f_{n-1}(f_{n-2}(...(f_{m}\left( s\right)
)...)),\;n\geq m+1,  \notag\end{aligned}$$and denote $$\eta _{j+1}:=\frac{f_{j}^{\prime \prime }\left( 1\right) }{\left(
f_{j}^{\prime }\left( 1\right) \right) ^{2}},\,b_{m}:=\frac{1}{2}\sum_{j=0}^{m-1}\eta _{j+1}e^{-S_{j}}$$(noting that both are positive quantities), then Assumption $A3$ implies (see, for instance, [@GK00])that for $m=0,1,....$ and $s\in[0,1]$, $$\frac{1}{1-f_{0,m}\left( s\right) }=\frac{e^{-S_{m}}}{1-s}+b_m.
\label{FracExp}$$

Later on we need various probability measures specified on the measurable space $\left( \Omega ,\mathcal{F}\right) .$ To distinguish them we use the symbols $\mathbf{E}$ and $\mathbf{P}$ to denote the expectation and probability law generated by the initial measure on the tuples $$(f_{0},f_{1},...,f_{n},\ldots ;Z_{0},Z_{1},...,Z_{n},\ldots )$$(there may be some occasional mild abuse of this notation however we do not anticipate that it will lead to confusion) and the symbols $\mathbf{E}_{\;\mathbf{k}},\;\mathbf{P}_{\;\mathbf{k}}\;$ to denote the expectation and probability law under the fixed environment  $\mathbf{\bar{k}}=\left( 
\mathbf{k}_{0},\mathbf{k}_{1},...,\mathbf{k}_{n},...\right) $. Along with the basic probability space $\left( \Omega ,\mathcal{F},\mathbf{P}\right) $ we deal with two its copies $\Big(\Omega 