n\right) <nt\right) =0.  \label{min42}\end{aligned}$$

\[EstO\]Assume that $A1-$ $A2$ hold. For any $R\in \mathbb{Z}$, any $t\in (0,1),$ and $N\in \left\{ n-1,n\right\} $ $$\left\{ O_{\tau (nt,n)+R,N}|\tau (n)<nt\right\} \overset{d}{\rightarrow }\frac{\theta _{R}}{1-\theta _{R}}.$$

**Proof**. We have$$O_{\tau (nt,n)+R,N}=\frac{1}{\alpha _{N}(\tau (nt,n)+R)}\times \frac{f_{\tau
(nt,n)+R,N}(0)}{1-f_{\tau (nt,n)+R,N}(0)}-f_{\tau (nt,n)+R,N}(0)\beta
_{N}(\tau (nt,n)+R).  \label{uu0}$$By Lemmas \[Lprospect1\] and \[Lprospect2\]$$\frac{1}{\alpha _{N}(\tau (nt,n)+R)}\times \frac{f_{\tau (nt,n)+R,N}(0)}{1-f_{\tau (nt,n)+R,N}(0)}\overset{d}{\rightarrow }\frac{\theta _{R}}{1-\theta _{R}},\,n\rightarrow \infty .  \label{uu1}$$Further, by (\[min42\])$$\begin{aligned}
\lefteqn{\beta _{N}(\tau (nt,n)+R)I\left\{ \tau (n)<nt\leq \tau
(nt,n)+R\right\} } \\
&\leq &\beta _{N}(nt)I\left\{ \tau (n)<nt\leq \tau (nt,n)+R\right\}   \notag
\\
&\leq &\beta _{N}(nt)I\left\{ \tau (n)<nt\right\} \overset{p}{\rightarrow }0
\label{uu2}\end{aligned}$$as $n\rightarrow \infty $. Using (\[uu1\]) and (\[uu2\]) to evaluate (\[uu0\]) proves the lemma.$\square $

\[Lpreelimit1\]Assume that $A1-$ $A2$ hold. For any $t\in \left(
0,1\right) ,$ any fixed $R\in \mathbb{Z}$, and $\varepsilon >0$$$\limsup_{n\rightarrow \infty }\mathbf{P}\left( \beta _{n}\left( \tau
(nt)+R\right) <1-\varepsilon ;\tau \left( n\right) \geq nt\right) =0.$$

**Proof**. Since $$e^{S_{m}}\left( 1-f_{m,n}\left( s\right) \right) \leq e^{S_{m+1}}\left(
1-f_{m+1,n}\left( s\right) \right)$$for any $m<n,$ the elements of the sequence $\left\{ \beta _{n}(m),0\leq
m\leq n\right\} $ are monotone decreasing in $m$ for any fixed $n.$ On the other hand, for any $\delta >0$ there exists $\varepsilon _{1}>0$ such that$$\mathbf{P}\left( \tau (nt)+R>nt(1-\varepsilon _{1})\right) <\delta$$for all $n\geq n_{0}=n_{0}(\delta ,\varepsilon _{1}).$ Thus, we have for all $n\geq n_{0}$ $$\begin{aligned}
\lefteqn{\mathbf{P}\left( \beta _{n}\left( \tau (nt)+R\right) <1-\varepsilon
