^{-},\mathcal{F}^{-},\mathbf{P}^{-}\Big)$ and $\Big(\Omega ^{+},\mathcal{F}^{+},\mathbf{P}^{+}\Big)$. Denote by $\left\{ f_{n}^{-},n\geq 0\right\} $ and $\left\{ f_{n}^{+},n\geq 0\right\} $ two sequences of the random environment and by $\left\{ S_{n}^{-},n\geq
0\right\} $ and $\left\{ S_{n}^{+},n\geq 0\right\} $ the corresponding associate random walks specified on $\Big(\Omega ^{-},\mathcal{F}^{-},\mathbf{P}^{-}\Big)$ and $\Big(\Omega ^{+},\mathcal{F}^{+},\mathbf{P}^{+}\Big),$ respectively. Later on any characteristics or random variables related with $\left\{ f_{n}^{-},n\geq 0\right\} $ and $\left\{
f_{n}^{+},n\geq 0\right\} $ are superscripted with the symbols $-$ or $+$, respectively. Following this practice, we write$$\Gamma ^{-}=\min \{n\geq 1:S_{n}^{-}\geq 0\}$$and $$\gamma ^{+}=\min \{n\geq 1:S_{n}^{+}<0\}.$$We also study various properties of the pair of branching processes in random environment given the event $$\mathcal{A}_{k,p}:=\left\{ \Gamma ^{-}>k,\gamma ^{+}>p\right\} .
\label{doublegamma}$$

Set $D=\sum_{j=1}^{\infty }\mathbf{P}(S_{j}=0)$. In addition to the measures $\mathbf{P}^{-}$ and $\mathbf{P}^{+}$ we define measures $\mathbf{\hat{P}}^{-} $ and $\mathbf{\hat{P}}^{+}$ on $\Big(\Omega ^{-},\mathcal{F}^{-}\Big)$ and $\Big(\Omega ^{+},\mathcal{F}^{+}\Big)$ $\ $whose restrictions $\mathbf{\hat{P}}_{k}^{-}$ and $\mathbf{\hat{P}}_{l}^{+}$ on the $\sigma $-algebras $\mathcal{F}_{k}^{-}$ and $\mathcal{F}_{p}^{+},\,k,p\in \mathbb{N}$ are specified by $$\begin{aligned}
\mathbf{\hat{P}}_{k}^{-}(\mathcal{A}^{-}) &=&e^{D}\int_{\mathcal{A}^{-}}U\left( -S_{k}^{-}\right) I\{\Gamma ^{-}>k\}d\mathbf{P}^{-},\,\mathcal{A}^{-}\in \mathcal{F}_{k}^{-},  \label{defMes-} \\
\mathbf{\hat{P}}_{p}^{+}(\mathcal{A}^{+}) &=&\int_{\mathcal{A}^{+}}V(S_{p}^{+})I\{\gamma ^{+}>p\}d\mathbf{P}^{+},\,\mathcal{A}^{+}\in 
\mathcal{F}_{p}^{+}.  \label{defMes+}\end{aligned}$$One can check (see, [@VD05]) that the sequences $\left\{ \mathbf{\hat{P}}_{k}^{-},\,k\in \mathbb{N}\right\} ,$ $\left\{ \mathbf{\hat{P}}_{p}^{-},\,p\in \mathbb{N}\right\} $ consist of well-defined and consistent probability measures. The probabilistic sense of these measures is rather transparent: the restriction of $\mathbf{\hat{P}}^{-}$ to $\mathcal{F}_{k}^{-}$ is 