\frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\left\vert \tau (n)\geq
nt\right. \right\} \rightarrow \zeta$$and$$\left\{ \frac{1-f_{nt,n}(0)}{e^{S_{\tau (nt,n)}-S_{nt}}}\left\vert \tau
(n)<nt\right. \right\} \rightarrow \zeta$$where $\zeta $ is defined in (\[aszet\]).

**Proof**. Our arguments follow the same line of reasoning as the proof of Lemma \[Lprelim4\]. We have$$\begin{aligned}
\lefteqn{\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\right\} ;\tau (n)\geq nt\right] } \\
&=&\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau
(nt)}}}\right\} \right]  \\
&&\hspace{2cm}-\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\right\} ;\tau (n)<nt\right] .\end{aligned}$$Introducing once again two independent environmental sequences $\left\{
f_{n}^{-},n\geq 0\right\} $ and $\left\{ f_{n}^{+},n\geq 0\right\} ,$ setting$$T_{\tau (nt),nt}^{\ast }:=\exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\right\} ,T_{l,nt-l}:=\exp \left\{ -\lambda \frac{1-f_{l,0}^{-}(f_{0,nt-l}^{+}(0))}{e^{S_{l}^{-}}}\right\}$$and observing that by (\[aszet\])$$\frac{1-f_{l,0}^{-}(f_{0,nt-l}^{+}(0))}{e^{S_{l}^{-}}}\rightarrow \zeta
^{-}(q^{+})=\zeta$$$\mathbf{\hat{P}}$ a.s. as $\min (l,nt-l)\rightarrow \infty ,$  we conclude by Lemma \[Cfran\] that for any $\lambda \in \left( 0,\infty \right) $$$\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\right\} \right] \rightarrow \mathbf{\hat{E}}\left[ \exp \left\{ -\lambda
\zeta \right\} \right]$$as $n\rightarrow \infty $. Further, $$\begin{aligned}
\lefteqn{\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau (nt)}}}\right\} ;\tau (n)<nt\right] } \\
&=&\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,nt}(0)}{e^{S_{\tau
(n)}}}\right\} ;\tau (n)<nt\right]  \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,j}(f_{j,nt}(0))}{e^{S_{j}}}\right\} ;\tau (n)=j\right]  \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{0,j}(f_{j,nt}(0))}{e^{S_{j}}}\right\} ;\min_{0\leq
r<j}S_{r}>S_{j},\min_{j+1\leq r\leq n}S_{r}\geq S_{j}\right]  \\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{j,0}^{-}(f_{0,nt-j}^{+}(0))}{e^{S_{j}^{-}}}\right\} ;A_{j,n-j}\right] 
\\
&=&\sum_{j=0}^{nt-1}\mathbf{E}\left[ \exp \left\{ -\lambda \frac{1-f_{j,0}^{-}(f_{0,nt-j}^{+}(0))}{e^{S_{j}^{-}}}\right\} \left\vert
A_{j,n-j}\right. \right] \mathbf{P}\left( \tau (n)=j\right) .\end{aligned}$$By (\[aszet\]) $$\frac{1-f_{j,0}^{-}(f_{0,nt-j}^{+}(0))}{e^{S_{j}^{-}}}\rightarrow \zeta$$$\mathbf{\hat{P}}$ a.s. as $\min 