## 3 Proposed Strategy
### Approximation Error Analysis  
We respectively define the judgment result of every local detector and the FC as \(X_{i}\), then \(W\) can be written as it is shown in Equation (16):  
\[W=\sum_{i=1}^{K}X_{i}, \tag{16}\]  
where \(X_{i}\) is the Bernoulli random variable. Equations (17)-(20) are defined as follows:  
\[P(X_{i}=1)=1-P(X_{i}=0)=p_{i}, \tag{17}\]  
\[\lambda=\sum_{i=1}^{K}p_{i}, \tag{18}\]  
\[W^{i}=\sum_{k\
eq i}X_{k}, \tag{19}\]  
\[V^{i}=\sum_{|k-i|>m}X_{k}, \tag{20}\]  
where \(m\) denotes the nonnegative integer. For every real-valued function \(f\) defined on the non-negative integers [34], Equations (21)-(24) is obtained:  
\[ \begin{array}{l}E[Wf(W)-\lambda f(W+1)]=\sum_{i=1}^{n}E[X_{i}f(W^{i}+1)]- \lambda E[f(w+1)]\\ =\ \sum_{i=1}^{n}E\big{\{}\,X_{i}[f(W^{i}+1)-f(V^{i}+1)]\big{\}}+\sum_{i=1}^{n}E [(X_{i}-p_{i})f(V^{i}+1)]+\sum_{i=1}^{n}E\big{\{}\,p_{i}[f(V^{i}+1)-f(W+1)] \big{\}}.\end{array}  \tag{21}\]  
We define:  
\[Y_{ij}=V^{i}+\sum_{k=i-m}^{j}X_{k}, \tag{22}\]  
\[Y_{ij}^{\#}=V^{i}+\sum_{k=i-m,k\
eq i}^{j}X_{k}, \tag{23}\]  
\[\Delta f(w)=f(w+1)-f(w). \tag{24}\]  
Then, Equation (21) can be reformulated as it is shown in Equation (25):  
\[E[Wf(W)-\lambda f(W+1)]=\sum\sum_{\lambda<|i-j|\leq m}E[X_{i}X_{j}\Delta f(Y _{ij-1}^{\#}+1)]+\sum_{i=1}^{n}E[(X_{i}-p_{i})f(V^{i}+1)]-\sum\sum_{0<|i-j|\leq m }p_{i}E[X_{j}\Delta f(Y_{ij-1}+1)]. \tag{25}\]  
To make this approximation more apparent, we define Equation (26) as follows:  
\[ \begin{array}{l}wf(w)-\lambda f(w+1)=h(w)-\phi_{\lambda}h,\\ \phi_{\lambda}h=e^{-\lambda}\sum_{k=0}^{\infty}h(k)\lambda^{k}/k!,\end{array}  \tag{26}\]  
where \(h\) is a bounded real-valued function on non-negative integers. The solution of Equation (24) can be given by Equation (27):  
\[f(w)=-(w-1)!\lambda^{-w}\!\!\sum_{k=0}^{w-1}[h(k)-e^{-\lambda}\sum_{k=0}^{ \infty}h(k)\lambda^{k}/k!]\lambda^{k}/k!=(w-1)!\lambda^{-w}\!\!\sum_{k=w}^{ \infty}[h(k)-e^{-\lambda}\!\!\sum_{k=0}^{\infty}h(k)\lambda^{k}/k!]\lambda^{k} /k!. \tag{27}\]  
We denote the solution of Equation (26) for \(w\geq 1\) by \(S_{\lambda}h(w)\). According to Equations (21) and (27), Equation (28) can be obtained:  
\[ \begin{array}{l}Eh(W)=e^{-\lambda}\!\!\sum_{k=0}^{\infty}h(k)\lambda^{k}/k! +\sum\!\sum_{0<|i-j|\leq m}E[X_{i}X_{j}\Delta S_{\lambda}h(Y_{ij-1}^{\#}+1)] \\ +\!\sum_{i=1}^{n}E[(X_{i}-p_{i})S_{\lambda}h(V^{i}+1)]-\sum\!\sum_{|i-j|\leq m }p_{i}E[X_{j}\Delta S_{\lambda}h(Y_{ij-1}+1)].\end{array}  \tag{28}\]  
**Theorem 1. ** If \(X_{1},\ \cdots,\ X_{n}\) are independent and \(\widetilde{p}=\max_{1\leq i\leq n}p_{i}\leq\lambda/2\), then for \(|h|\leq 1\) and \(n\geq 2\), Equation (29) is obtained:  
\[\left|\,Eh(W)-\phi_{\lambda}h+(\sum_{i=1}^{n}p_{i}^{2})\phi_{\lambda}U_{ \lambda}h\right|\leq C\lambda^{-1}\!\sum_{i=1}^{n}p_{i}^{2}, \tag{29}\]  
where \(C\) a constant [31]. **Proof. ** According to Equation (22), Equations (30) and (31) is obtained:  
\[Eh(W)=\phi_{\lambda}h-nEp_{I}^{2}E^{I}U_{\lambda}h(W^{\#}), \tag{30}\]  
\[W^{\#}=\sum_{i\
eq I}X_{i}, \tag{31}\]  
where \(I\) and \(J\) are two random indices. \(\Box\)  
Equation (30) can be further denoted as it is shown in Equations (32)-(34):  
\[Eh(W)=\phi_{\lambda}h-E_{I}^{2}p_{I}^{2}[\phi_{\lambda}U_{\lambda}h-(n-1)E^{I }p_{I}^{2}U_{\lambda}U_{\lambda}h(W^{\#})]=\phi_{\lambda}h-(\sum_{i=1}^{n}p_{ i}^{2})\phi_{\lambda}U_{\lambda}h-nEp_{I}^{2}(\phi_{\lambda}U_{\lambda}h-\phi_{ \lambda}U_{\lambda}h)+n(n-1)Ep_{I}^{2}p_{I}^{2}E^{I}U_{\lambda}U_{\lambda}U_{ \lambda}h(W^{\#}), \tag{32}\]\[\lambda^{*} =\sum_{i\
eq I}p_{i}, \tag{33}\] \[W^{\#\#} =\sum_{i\
eq I,J}X_{i}. \tag{34}\]  
Based on the Lemmas 5.2 and 5.3 in [31], Equation (32) can be written as it is shown in Equation (35):  
\[ \begin{split}|\phi_{\lambda^{*}}U_{\lambda}h-\phi_{\lambda}U_{ \lambda}h|&\leq p_{I}[\phi_{\lambda}|LU_{\lambda}h|+\phi_{ \lambda^{*}}|U_{\lambda}h|]\\ &\leq 24p_{I}\lambda^{-1}.\end{split}  \tag{35}\]  
According to Lemma 5.5 in [34], Equations (36) and (37) can be obtained:  
\[