f_{j+R,N}(0)};\min_{0\leq
r<j}S_{r}>S_{j},\min_{j+1\leq r\leq n}S_{r}\geq S_{j}\right]  \\
&=&\sum_{j=nt}^{n}\mathbf{E}\left[ e^{-\lambda
f_{R,0}^{-}(f_{0,N-j}^{+}(0))};A_{j,n-j}\right]  \\
&=&\sum_{j=nt}^{n}\mathbf{E}\left[ e^{-\lambda
f_{R,0}^{-}(f_{0,N-j}^{+}(0))}|A_{j,n-j}\right] \mathbf{P}(\tau (n)=j)\end{aligned}$$and since $f_{R,0}^{-}(f_{0,N-j}^{+}(0))\rightarrow f_{R,0}^{-}(q^{+})$ $\mathbf{\hat{P}}-$a.s as $n-j\rightarrow \infty $ we have by Lemma [Lfran]{} that $$\lim_{n-j\rightarrow \infty }\mathbf{E}\left[ e^{-\lambda
f_{R,N-j}(0)}|A_{j,n-j}\right] =\mathbf{\hat{E}}\left[ e^{-\lambda
f_{R,0}^{-}(q^{+})}\right] .$$This and the generalized arcsine law give$$\lim_{n\rightarrow \infty }\mathbf{E}\left[ e^{-\lambda f_{\tau
(nt,n)+R,N}(0)};\tau (n)\geq nt\right] =\mathbf{\hat{E}}\left[ e^{-\lambda
f_{R,0}^{-}(q^{+})}\right] \mathbf{P}\left( \tau \geq t\right) .$$

Thus,$$\lim_{n\rightarrow \infty }\mathbf{E}\left[ e^{-\lambda f_{\tau
(nt,n)+R,N}(0)I\left\{ \mathbf{\tau }(nt,n)+R\geq 0\right\} };\tau (n)<nt\right] =\mathbf{\hat{E}}\left[ e^{-\lambda f_{R,0}^{-}(q^{+})}\right] 
\mathbf{P}\left( \tau \leq t\right)$$proving the lemma for $R<0$.

The case $R\geq 0$ can be treated in a similar way by observing that, in this case, for all $j\leq N-R$ $$\mathbf{E}\left[ e^{-\lambda f_{\tau (n)+R,N}(0)};\tau (n)=j\right] =\mathbf{E}\left[ e^{-\lambda f_{R,N-j}^{+}(0))}|A_{j,n-j}\right] \mathbf{P}(\tau
(n)=j)$$and that (\[asq\]) holds.$\square $

In what follows we need some properties of the random variable $\beta
_{n}(m) $.

\[min22\]([@VD3], Lemma 17) Assume that $A1-$ $A2$ hold. For any $\varepsilon >0$ and $N\in \left\{ n-1,n\right\} $$$\limsup_{R,n\rightarrow \infty }\mathbf{P}\left( \beta _{N}\left( \tau
\left( n\right) +R\right) >\varepsilon \right) =0.  \label{betvan}$$

\[min4\]([@VD3], Corollary 4) Assume that $A1-$ $A2$ hold. For any $t\in \left( 0,1\right) $ and $\varepsilon >0$$$\begin{aligned}
&&\limsup_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\left( nt\right)
<1-\varepsilon ;\tau \left( n\right) \geq nt\right) =0,  \label{min41} \\
&&\limsup_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\left( nt\right)
>\varepsilon ;\tau \left( 