;\tau \left( n\right) \geq nt\right) } \\
&\leq &\delta +\mathbf{P}\left( \beta _{n}\left( nt(1-\varepsilon
_{1})\right) <1-\varepsilon ;\tau \left( n\right) \geq nt\right) \\
&\leq &\delta +\mathbf{P}\left( \beta _{n}\left( nt(1-\varepsilon
_{1})\right) >1-\varepsilon ;\tau \left( n\right) \geq nt(1-\varepsilon
_{1})\right) .\end{aligned}$$To complete the proof it remains to recall (\[min41\]) .$\square $

\[Lprelim3\]Assume that $A1-$ $A2$ hold. For any $R\in \mathbb{Z}$, any $t\in (0,1),$ and $N\in \left\{ n-1,n\right\} $ $$\left\{ f_{\tau (nt)+R,N}(0)|\tau (n)\geq nt\right\} \overset{d}{\rightarrow 
}1,\,n\rightarrow \infty .$$

**Proof**. For a fixed $\varepsilon \in (0,1)$ introduce the event $$\mathcal{H}_{n}(\varepsilon )=\left\{ \omega :\left\{ \tau (nt),\tau
(nt)+R,\tau (n)\right\} \cap \lbrack nt(1-\varepsilon ),nt(1+\varepsilon
)]=\oslash \right\}$$and let $\mathcal{\bar{H}}_{n}(\varepsilon )$ be the complement of  $\mathcal{H}_{n}(\varepsilon ).$ By the generalized arcsine law $$\begin{aligned}
\lim_{\varepsilon \downarrow 0}\lim_{n\rightarrow \infty }\mathbf{P}\left( 
\mathcal{\bar{H}}_{n}(\varepsilon )\right)  &\leq &\lim_{\varepsilon
\downarrow 0}\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (nt)\in
\lbrack nt(1-\varepsilon ),nt(1+\varepsilon )]\right)  \\
&&+\lim_{\varepsilon \downarrow 0}\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (nt)+R\in \lbrack nt(1-\varepsilon ),nt(1+\varepsilon )]\right) 
\\
&&+\lim_{\varepsilon \downarrow 0}\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (n)\in \lbrack nt(1-\varepsilon ),nt(1+\varepsilon )]\right)  \\
&=&0.\end{aligned}$$Hence, to prove the lemma it is suffices to show that $$\left\{ f_{\tau (nt)+R,N}(0)|\mathcal{H}_{n}(\varepsilon );\tau (n)\geq
nt\right\} \overset{d}{\rightarrow }1,\,n\rightarrow \infty .$$Clearly,$$\begin{aligned}
\lefteqn{\left( 1-f_{\tau (nt)+R,N}(0)\right) I\left\{ \mathcal{H}_{n}(\varepsilon );\tau (n)\geq nt\right\} } \\
&\leq &\left( 1-f_{\tau (nt)+R,\tau (n)}(0)\right) I\left\{ \mathcal{H}_{n}(\varepsilon );\tau (n)>nt(1+\varepsilon )\right\}  \\
&\leq &e^{S_{\tau (n)}-S_{\tau 