s\frac{f_{0,m}^{\prime }(sf_{m,n}(0))}{f_{0,m}^{\prime }(f_{m,n-1}(0))}.$$It is easy to conclude from (\[FracExp\]) that under Assumption $A3$$$f_{0,m}^{\prime }(s)=\frac{e^{-S_{m}}}{\left( 1-s\right) ^{2}}(1-f_{0,m}(s))^{2}.$$Therefore,$$\frac{f_{0,m}^{\prime }(sf_{m,n}(0))}{f_{0,m}^{\prime }(f_{m,n-1}(0))}=\frac{(1-f_{0,m}(sf_{m,n}(0)))^{2}}{\left( 1-sf_{m,n}(0)\right) ^{2}}\frac{\left(
1-f_{m,n-1}(0)\right) ^{2}}{(1-f_{0,n-1}(0))^{2}}.$$Further, again making use of (\[FracExp\]), we have for $s\in[0,1]$ $$\begin{aligned}
\frac{1-f_{0,m}(sf_{m,n}(0))}{1-sf_{m,n}(0)} &=&\frac{1}{e^{-S_{m}}+\left(
1-sf_{m,n}(0)\right) b_{m}} \\
&=&\frac{1}{e^{-S_{m}}+\left( 1-f_{m,n}(0)\right) b_{m}+(1-s)f_{m,n}(0)b_{m}}
\\
&=&\frac{1}{(1-f_{m,n}(0))\left( 1-f_{0,n}(0)\right)
^{-1}+(1-s)f_{m,n}(0)b_{m}} \\
&=&\frac{1-f_{0,n}(0)}{1-f_{m,n}(0)}\times \frac{1}{1+(1-s)O_{m,n}}\end{aligned}$$where the third equality follows from the first equality when $s=1$. As a result we have$$\frac{f_{0,m}^{\prime }(sf_{m,n}(0))}{f_{0,m}^{\prime }(f_{m,n-1}(0))}=\frac{1}{1+(1-s)O_{m,n}}\Delta _{m,n}.$$Similarly,$$\frac{f_{0,m}^{\prime }(sf_{m,n-1}(0))}{f_{0,m}^{\prime }(f_{m,n}(0))}=\frac{1}{1+(1-s)O_{m,n-1}}\Delta _{m,n}^{-1}.$$The lemma is proved.$\square$

To proceed further we need to formulate for future reference several known statements. In the next two lemmas recall that $\mathcal{A}_{n,r}$ was defined in (\[doublegamma\]).

\[Lfran\] ([@VD3], Lemma 3) Let Assumption $A1$ be valid and let $T_{l,p},\,l,p\in \mathbb{N},$ be a tuple of uniformly bounded random variables such that, for any pair $l,p$ the random variable $T_{l,p}$ is measurable with respect to the $\sigma $-algebra $\mathcal{F}_{l}^{-}\times 
\mathcal{F}_{p}^{+}$ . Then $$\lim_{\min (n,r)\rightarrow \infty }\mathbf{E}[T_{l,p}\,|\,\mathcal{A}_{n,r}]=\hat{\mathbf{E}}[T_{l,p}].  \label{g11}$$More generally, if the tuple $\{T_{n,r},\,n,r\in \mathbb{N}\}$ consists of the uniformly bounded random variables which are adopted to the flow of the $\sigma -$algebras$\{\mathcal{F}_{n}^{-}\times \mathcal{F}_{r}^{+}\}_{n\geq 1,r\geq 1},$ and $\lim_{\min (n,r)\rightarrow \infty
}T_{n,r}=:T$ exists $\hat{\mathbf{P}}-$ a.s. then $$\lim_{\min (n,r)\rightarrow \infty }\mathbf{E}[T_{n,r}\,|\,\mathcal{A}_{n,r}]=\hat{\mathbf{E}}[T].  \label{g22}$$

\[Cfran\] ([@VD3], Lemma 4) Let Assumption $A1$ be valid and let $T$ and $T_{l,p},\,l,p\in \mathbb{N}$ be 