a tuple of random variables meeting the conditions of Lemma \[Lfran\]. If $T_{l,p}^{\ast },\,l,p\in \mathbb{N}$ is a tuple of uniformly bounded random variables such that for any pair $l,p$ the random variable, $T_{l,p}^{\ast }$ is measurable with respect to the $\sigma $-algebra $\mathcal{F}_{l}^{-}\times \mathcal{F}_{p}^{+}$ and $$\mathbf{E}[T_{\tau (n),n}^{\ast }\,|\,\tau (n)=l]=\mathbf{E}[T_{l,n-l}\,|\,\mathcal{A}_{l,n-l}],$$then $$\lim_{n\rightarrow \infty }\mathbf{E}[T_{\tau (n),n}^{\ast }]=\hat{\mathbf{E}}[T].  \label{g112}$$

** **

For $0\leq m\leq n$ set $$\alpha _{n}(m):=\frac{1-f_{0,m}(0)}{1-f_{0,n}(0)},\qquad \beta _{n}(m):=\frac{1-f_{0,n}\left( 0\right) }{e^{S_{m}}\left( 1-f_{m,n}\left( 0\right)
\right) }.\,  \label{alfbet}$$Clearly, $\alpha _{n}(m)\geq 1,\ \beta _{n}(m)\leq 1,$ and, in addition,$$\begin{aligned}
O_{m,n} &=&\frac{1}{\alpha _{n}(m)}\times \frac{f_{m,n}(0)}{1-f_{m,n}(0)}-f_{m,n}(0)\beta _{n}(m)  \notag \\
&=&f_{m,n}(0)\beta _{n}(m)\left( \frac{e^{S_{m}}}{1-f_{0,m}(0)}-1\right) .
\label{DDoubleRepres}\end{aligned}$$

\[min1\] ([@VD3], Lemma 16) Assume that $A1-$ $A2$ hold. Then for $R\in \mathbb{Z}$ and any $\varepsilon >0$ $$\limsup_{R,n\rightarrow \infty }\mathbf{P}\left( \alpha _{n}(\tau
(n)+R)>1+\varepsilon \right) =0.$$

Since the random variable $\alpha _{n}(\tau (n)+R)$ is not defined for $\tau
(n)+R<0$ or $\tau (n)+R>n$ we should formally write the statement of the lemma as $$\limsup_{R,n\rightarrow \infty }\mathbf{P}\left( \alpha _{n}(\tau
(n)+R)>1+\varepsilon ;n\geq \tau (n)+R\geq 0\right) =0$$However, by the generalized arcsine law for each fixed $R\in Z$$$\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (n)+R\notin \lbrack
0,n]\right) =0.  \label{Inter1}$$For this reason here and in what follows we agree to treat $\alpha _{n}(\tau
(n)+R)$ as $\alpha _{n}(0)$ if $\tau (n)+R<0$ and $\alpha _{n}(n)$ if $\tau
(n)+R>n.$ 