have$$\begin{aligned}
\lefteqn{\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau (nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} ;\tau (n)\geq nt\right] } \\
&=&\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau (nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} \right] \\
&&\hspace{2cm}-\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau
(nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} ;\tau (n)<nt\right] .\end{aligned}$$Introducing once again two independent environmental sequences $\left\{
f_{n}^{-},n\geq 0\right\} $ and $\left\{ f_{n}^{+},n\geq 0\right\} ,$ setting$$T_{\tau (nt),nt}^{\ast }:=\exp \left\{ -\lambda \frac{1-f_{0,\tau (nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} ,T_{l,nt-l}:=\exp \left\{ -\lambda \frac{1-f_{l,0}^{-}(f_{0,R}^{+}(0))}{e^{S_{l}^{-}}e^{S_{R}^{+}}}\right\}$$and observing that$$\frac{1-f_{l,0}^{-}(f_{0,R}^{+}(0))}{e^{S_{l}^{-}}e^{S_{R}^{+}}}\rightarrow 
\frac{\zeta ^{-}(f_{0,R}^{+}(0))}{e^{S_{R}^{+}}}$$$\mathbf{\hat{P}}$ a.s. as $\min (l,n(1-t)-l)\rightarrow \infty ,$ we conclude by Lemma \[Cfran\] that for any $\lambda \in \left( 0,\infty
\right) $$$\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau (nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} \right] \rightarrow \mathbf{\hat{E}}\left[ \exp
\left\{ -\lambda \frac{\zeta ^{-}(f_{0,R}^{+}(0))}{e^{S_{R}^{+}}}\right\} \right]$$as $n\rightarrow \infty .$ Further, the same as in Lemma \[Lprelim3\] $$\begin{aligned}
\lefteqn{\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau (nt)+R}(0)}{e^{S_{\tau (nt)+R}}}\right\} ;\tau (n)<nt\right] } \\
&=&\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,\tau (n)+R}(0)}{e^{S_{\tau (n)+R}}}\right\} ;\tau (n)<nt\right] \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,j+R}(0)}{e^{S_{j+R}}}\right\} ;\tau (n)=j\right] \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,j+R}(0)}{e^{S_{j+R}}}\right\} ;\min_{0\leq
r<j}S_{r}>S_{j},\min_{j+1\leq r\leq n}S_{r}\geq S_{j}\right] \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{j,0}^{-}(f_{0,R}^{+}(0))}{e^{S_{j}^{-}}e^{S_{R}^{+}}}\right\} ;A_{j,n-j}\right] \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{j,0}^{-}(f_{0,R}^{+}(0))}{e^{S_{j}^{-}}e^{S_{R}^{+}}}\right\}
\left\vert A_{j,n-j}\right. \right] \mathbf{P}\left( \tau (n)=j\right) .\end{aligned}$$By (\[aszet\]) $$\frac{1-f_{j,0}^{-}(f_{0,R}^{+}(0))}{e^{S_{j}^{-}}e^{S_{R}^{+}}}\rightarrow 
\frac{\zeta ^{-}(f_{0,R}^{+}(0))}{e^{S_{R}^{+}}}$$$\mathbf{\hat{P}}$ a.s. as $\min (j,n-j)\rightarrow \infty .$ Hence we see by Lemma \[Lfran\] that $$\lim_{\min (j,n-j)\rightarrow \infty }\mathbf{E}\left[ \exp \left\{ -\lambda 
\frac{\zeta ^{-}(f_{0,R}^{+}(0))}{e^{S_{R}^{+}}}\right\} \left\vert
A_{j,n-j}\right. \right] =\mathbf{\hat{E}}\left[ \exp \left\{ 