concentrated only on the realizations of the environment whose associated random walks are negative for the first $k$ steps (except the starting point) while the restriction of $\mathbf{\hat{P}}^{+}$ to $\mathcal{F}_{p}^{+}$ is concentrated only on the realizations of the environment whose associated random walks are nonnegative for the first $p$ steps. Indeed, in an appropriate sense, $\mathbf{\hat{P}}^+$ and $\mathbf{\hat{P}}^- $ may be thought of as the random walks $S^+$ and $S^-$ conditioned to stay positive. See for example [@BD].

Further, on the measurable space $\Big(\Omega ^{-}\times \Omega ^{+},\mathcal{F}^{-}\times \mathcal{F}^{+}\Big)$ we specify the probability measure $\mathbf{\hat{P}}:=\mathbf{\hat{P}}^{-}\times \mathbf{\hat{P}}^{+},$ whose projection on the elements of the $\sigma $-algebra $\mathcal{F}_{k}^{-}\times \mathcal{F}_{p}^{+}$ is given by $$\mathbf{\hat{P}}(\mathcal{A})=e^{D}\int_{\mathcal{A}}U\left(
-S_{k}^{-}\right) V(S_{p}^{+})I\{\mathcal{A}_{k,p}\}d(\mathbf{P}^{-}\times 
\mathbf{P}^{+}),\mathcal{A}\in \mathcal{F}_{k}^{-}\times \mathcal{F}_{p}^{+}
\label{defMes}$$(see [@VD05] for more detailed description of this measure).

With the notation above in hand we list for further references some results established in  [@VD3] before moving to our main results.

1)

:   Under Assumptions $A1-A2$ for any $R\in \mathbb{N}_{0}$ there exists the limit$$q_{R}^{+}:=\lim_{n\rightarrow \infty }f_{R,n}^{+}(0)<1\ \ \hat{\mathbf{P}}^{+}\text{- a.s.}  \label{asq}$$(later on we write for brevity $q^{+}$ for $q_{0}^{+}$);

2)

:   the tuple of random functions $$\zeta _{l,m}^{-}(s):=\frac{1-f_{l,m}^{-}(s)}{e^{S_{l}^{-}-S_{m}^{-}}},\ m\in 
    \mathbb{N}_{0},l\geq m+1,  \label{zetam}$$is such that $\hat{\mathbf{P}}-$ a.s. the limit$$\zeta _{\infty ,m}^{-}(s):=\lim_{l\rightarrow \infty }\zeta _{l,m}^{-}(s)
    \label{defzetam}$$exists and is 