(On a notational point, here and in the sequel we understand $nt$ as $[nt]$, the integer part of $nt$). To formulate our results we need to specify a number of characteristics related with associated random walks.

Let $$\gamma _{0}:=0,\quad \gamma _{j+1}:=\min (n>\gamma _{j}:S_{n}<S_{\gamma
_{j}})$$and $$\Gamma _{0}:=0,\quad \Gamma _{j+1}:=\min (n>\Gamma _{j}:S_{n}>S_{\Gamma
_{j}}),\,j\geq 0,$$be the strict descending and strict ascending ladder epochs of $\{S_{n},n\geq 0\}$. Introduce the functions $$\begin{aligned}
V(x) &:&=\sum_{j=0}^{\infty }\mathbf{P}(S_{\gamma _{j}}\geq -x),\quad
x>0,\quad V\left( 0\right) =1,\quad V\left( x\right) =0,\quad x<0, \\
U(x) &:&=1+\sum_{j=1}^{\infty }\mathbf{P}(S_{\Gamma _{j}}<x),\quad x>0,\quad
U\left( 0\right) =1,\quad U(x)=0,\quad x<0,\end{aligned}$$and set$$\Theta (a):=\frac{\sum_{j=a}^{\infty }j^{2}\mathbf{K}{(}\left\{ {j}\right\} {)}}{\left( \sum_{r=0}^{\infty }r\mathbf{K}{(}\left\{ {r}\right\} {)}\right)
^{2}},\quad a\in \mathbb{N}.$$

**Assumption $A2$**. There exist $\varepsilon _{0}>0$ and $a\in \mathbb{N}_{0}$ such that $$\begin{aligned}
&&\mathbf{E}(\log ^{+}\Theta (a))^{\frac{1}{\rho }+\varepsilon _{0}}<\infty
\quad \mbox{and}\quad \mathbf{E}[V(X)(\log ^{+}\Theta (a))^{1+\varepsilon
_{0}}]<\infty ,  \label{A21} \\
&&\mathbf{E}(\log ^{+}\Theta (a))^{\frac{1}{1-\rho }+\varepsilon
_{0}}<\infty \quad \mbox{and}\quad \mathbf{E}[U(-X)(\log ^{+}\Theta
(a))^{1+\varepsilon _{0}}]<\infty .  \label{A22}\end{aligned}$$

One can find in [@AGKV] and [@VD3] more details demonstrating the importance of these conditions.

Finally, we impose a (rather specific) condition on the form of the probability generating functions of the underlying branching process in random environment.

**Assumption** $A3$. The random offspring generating functions $f_{n}(s),\,n=0,1,...,$ are fractional-linear, i.e., they have with probability 1 the following form $$f_{n}(s)=r_{n}+(1-r_{n})\frac{q_{n}}{1-p_{n}s}  