## 2 Problem Formulation
### Parameter Estimation Algorithm and Its Convergence  
Select the subset \(O_{N,i}^{0}\) from \(O_{N,i}=\{i,n+i,\ldots,(L_{N}-1)n+i\}\) for \(i=1,2,\ldots,n\), ensuring that the number of elements in the subsets corresponding to different \(i\) is the same. Combine these subsets into a new set:  
\[Z_{N}=\bigcup_{i=1}^{n}O_{N,i}^{0}. \tag{22}\]  
Let the number of elements in \(Z_{N}\) be \(N_{0}\), and \(N_{0}/N\to\alpha,N\to\infty,\alpha\in(0,1)\). In the scenario where \(p\) is unknown, improve CSTCS as follows:  
\[\gamma_{k}^{e} = \left\{ \begin{array}{ll}1,&k\in Z_{N};\\ I_{\{\xi_{k}=1\}}\sum_{i=1}^{n}I_{\{\text{rem}(k,n)=\text{rem}(i,n)\}}I_{\{ \text{rem}(\omega_{k,i},m)=0\}},&k\in\overline{Z}_{N},\end{array} \right. \tag{23}\] \[\omega_{k,i} = \omega_{k-1,i}+s_{k}I_{\{\text{rem}(k,n)=\text{rem}(i,n),k\in \mathbb{Z}_{N}\}},\,\,\,i=1,2,\ldots,n, \tag{24}\]  
where \(\omega_{0,i}=0,\,\,i=1,2,\ldots,n\), and \(\overline{Z}_{N}\) represents the complement of \(Z_{N}\) within \(\{1,2,\ldots,N\}\). The unknown parameter \(\theta\) can be estimated by the following algorithm:  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\] \[\xi_{N,i} = \frac{m}{L_{N}-N_{0}/n}\sum_{k=1}^{N}\frac{\gamma_{k}I_{\{k}\in \mathcal{O}_{N,i}^{0}\}}{1-\widehat{p}_{N}},\,\,\,i=1,2,\ldots,n, \tag{26}\]\[\widehat{p}_{N} = 1-\frac{1}{N_{0}}\sum_{k=1}^{N}\gamma_{k}I_{\{k\in Z_{N}\}}, \tag{27}\]  
where \(\overline{O}_{N,i}^{0}\) denotes the complement of \(O_{N,i}^{0}\) within \(O_{N,i}\) for \(i=1,2,\ldots,n\). **Lemma 1** ([32]). : _If the sequences of random variables \(\{X_{k}\}\), \(\{Y_{k}\}\), \(\{Z_{k}\}\) satisfy \(X_{k}\stackrel{{\mathrm{d}}}{{\to}}X\), \(Y_{k}\stackrel{{\mathrm{P}}}{{\to}}\lambda_{1}\), and \(Z_{k}\stackrel{{\mathrm{P}}}{{\to}}\lambda_{2}\), then we have_  
\[X_{k}Y_{k}+Z_{k}\stackrel{{\mathrm{d}}}{{\to}}\lambda_{1}X+ \lambda_{2},\;\;k\to\infty,\]  
_where \(\lambda_{1},\lambda_{2}\) are finite constants._  
**Theorem 2**. : _Under the uncertain communication (5), consider the system (1) that introduces the improved CSTCS (23) and (24) and binary quantization (2). If Assumptions 1 and 2 are true, the estimate \(\widehat{\theta}_{N}\) derived from (25)-(27) strongly converges to the true value, i.e.,_  
\[\widehat{\theta}_{N}\to\theta,\;\;\text{as}\;N\to\infty,\;\;\text{w.p.1}.\]  
Proof. : From (13) and (14), we have  
\[\mathbb{E}(s_{k}) = \mathrm{P}(s_{k}=1)=F(C-\phi_{k}^{T}\theta), \tag{28}\] \[\mathbb{E}(\gamma_{k}^{d}) = 1-p. \tag{29}\]  
And then, by the Kolmogorov law of large numbers [31], one can obtain  
\[\frac{1}{L_{N}-N_{0}/n}\sum_{k=1}^{N}s_{k}I_{\{k\in\overline{O}_{ N,i}^{0}\}} \to F_{i},\;N\to\infty,\;\;i=1,2,\ldots,n, \tag{30}\] \[\frac{1}{L_{N}-N_{0}/n}\sum_{k=1}^{N}\gamma_{k}^{d}I_{\{k\in \overline{O}_{N,i}^{0}\}} \to 1-p,\;N\to\infty,\;\;i=1,2,\ldots,n,\] (31) \[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\]0  
When \(k\in Z_{N}\), \(\gamma_{k}^{t}=1\), and, thus, from (32) it follows that  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\]1  
From (23) and (24), we can know \(\frac{1}{L_{N}-N_{0}/n}\sum_{k=1}^{N}s_{k}I_{\{k\in\overline{O}_{N,i}^{0}\}}= \frac{m}{L_{N}-N_{0}/n}\sum_{k=1}^{N}\gamma_{k}^{t}I_{\{k\in\overline{O}_{N,i} ^{0}\}}+\frac{\overline{C}_{N,i}}{L_{N}-N_{0}/n}\), where  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\]2  
Noting that \(\overline{C}_{N,i}\in\{0,1,2,\ldots,m-1\}\), we have  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\]3  
According to (29), when \(k\in\overline{Z}_{N}\),  
\[\widehat{\theta}_{N} = \Phi^{-1}\Big{[}C-F^{-1}(\xi_{N,1}),\ldots,C-F^{-1}(\xi_{N,n}) \Big{]}^{T}, \tag{25}\]4Denote \(\widetilde{\gamma}_{k}=\gamma_{k}-(1-p)\gamma_{k}^{\epsilon}\).