conclude that $\mathbf{\hat{P}}$ a.s. $$\begin{aligned}
&&\lim_{\min (l.n-l)\rightarrow \infty }T_{l,n-l-1} \\
&=&\lim_{\min (l.n-l)\rightarrow \infty }\frac{1-f_{l,0}^{-}(f_{0,n-l-1}^{+}(f_{n-1}(0)))}{e^{S_{l}^{-}}}\times \frac{e^{S_{l}^{-}}}{1-f_{l,0}^{-}(f_{0,n-l-1}^{+}(0))}=1.\end{aligned}$$To finish the proof of (\[ratio1\]) it remains to observe that $$\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (n)\neq \tau (n-1)\right)
=0,$$to check that $$\mathbf{E}\left[ T_{\tau (n-1),n-1}^{\ast }|\tau (n-1)=l\right] =\mathbf{E}\left[ T_{l,n-l-1}|A_{l,n-l-1}\right] ,$$and to apply Lemma \[Cfran\].$\square $

**Proof of Theorem** \[Tclose\]. To prove the statement of the theorem it suffices to combine Lemmas \[EstOprel\], \[EstO\], and [DeltaVanish]{}.$\square$

Proof of Theorem \[Tfar\]
=========================

\[Lratio\]Under Assumptions $A1-A2$,$$\frac{O_{nt,n-1}}{O_{nt,n}}\overset{p}{\rightarrow }1\text{ \ as \ }n\rightarrow \infty .  \label{Oratio}$$

**Proof**. We have$$\frac{O_{nt,n-1}}{O_{nt,n}}=\frac{f_{nt,n-1}(0)}{f_{nt,n}(0)}\times \frac{1-f_{0,n-1}(0)}{1-f_{0,n}(0)}\times \frac{1-f_{nt,n}(0)}{1-f_{nt,n-1}(0)}.$$Since the process is critical,$$f_{nt,n}(0)\overset{d}{=}f_{0,n(1-t)}(0)\rightarrow 1\text{ \ a.s.}
\label{c1}$$as $n\rightarrow \infty .$ Similarly to (\[ratio1\]) $$\frac{1-f_{nt,n}(0)}{1-f_{nt,n-1}(0)}\overset{d}{=}\frac{1-f_{0,n(1-t)}(0)}{1-f_{0,n(1-t)-1}(0)}\overset{d}{\rightarrow }1  \label{c3}$$as $n\rightarrow \infty .$ Combining (\[ratio1\]), (\[c1\]), and ([c3]{}) gives (\[Oratio\]).$\square$

\[LOleftRight\]Under Assumptions $A1$ $-$ $A2$, for any $t\in (0,1)$ as $n\rightarrow \infty $$$\left\{ O_{nt,n}\left( 1-f_{nt,n}(0)\right) \left\vert \tau (n)<nt\right.
\right\} \overset{d}{\rightarrow }1,  \label{di1}$$and$$\left\{ O_{nt,n}\left( 1-f_{0,nt}(0)\right) e^{-S_{nt}}\left\vert \tau
(n)\geq nt\right. \right\} \overset{d}{\rightarrow }1.  \label{di2}$$

**Proof**. To prove (\[di1\]) it suffices to observe that $$O_{nt,n}\left( 1-f_{nt,n}(0)\right) =\frac{1}{\alpha _{n}(nt)}\times
f_{nt,n}(0)-f_{nt,n}(0)\beta _{n}(nt)\left( 1-f_{nt,n}(0)\right)$$and to apply Lemma \[min2\] and (\[min42\]), while to demonstrate ([di2]{}) one should write $$O_{nt,n}(1-f_{0,nt}(0))e^{-S_{nt}}=f_{nt,n}(0)\beta _{n}(nt)\left[ 1-\left(
1-f_{nt,n}(0))\right) e^{-S_{nt}}\right]$$and to use Lemma \[min4\] and (\[min41\]).$\square$

\[Comm1\]Under Assumptions $A1-$** **$A2$, for any $t\in (0,1)$ as $n\rightarrow \infty $$$\left\{ 