\right\} $ be the space of elementary events with$$\omega =\left\{ \mathbf{k}_{0},\mathbf{k}_{1},\ldots ;z(0),z(1),...\right\} ,$$$\mathcal{F}$ be the natural $\sigma $-algebra generated by the subsets of $\Omega $ and $\mathbf{P}$ be the corresponding probability measure on the $\left( \Omega ,\mathcal{F}\right) $. The triple $\left( \Omega ,\mathcal{F},\mathbf{P}\right) $ is be our basic probability space. By $\mathcal{F}_{n},n\geq 1,$ we denote the projection of $\mathcal{F}$ on $$\Omega _{n}:=\left\{ \omega _{n}=\left( \mathbf{k}_{0},\mathbf{k}_{1},\ldots
,\mathbf{k}_{n-1};z(0),z(1),...,z(n-1)\right) \right\}$$and by $\mathbf{P}_{n}$ the projection of $\mathbf{P}$ on $\mathcal{F}_{n}.$

Let $$f\overset{d}{=}f_{0},\ X_{k}:=\ln f_{k-1}^{\prime}\left( 1\right) ,\,\eta
_{k}:=f_{k-1}^{\prime\prime}\left( 1\right) \left( f_{k-1}^{\prime}\left(
1\right) \right) ^{-2},k\in \mathbb{N}=\{1,2,...\};$$$$X\overset{d}{=}X_{1};\,S_{0}:=0,\;S_{k}:=X_{1}+...+X_{k},\,k\geq 1.$$The sequence $\left\{ S_{k},k\geq 0\right\} $ is called the associated random walk of the corresponding branching process in random environment.

Let $$T:=\min \left\{ k:Z_{k}=0\right\}$$and $\tau \left( m,n\right) :=\min \{i\in \lbrack m,n]:\;S_{j}\geq
S_{i},\;j=m,m+1,\ldots ,n\}$ be the left–most point of minimum of the random walk $\left\{ S_{k},k\geq 0\right\} \;$ on the discrete time interval $[m,n].$ In particular we shall write $\tau (n):=\tau \left(
0,n\right) $ is the left–most point of minimum of the random walk on the discrete time interval $[0,n].$

Properties of branching processes in random environment are specified to a great extent by the properties of the associated random walk. One of the most important conditions we impose on the characteristics 