**4.2 Convergence of the iterative algorithm**

Now, we show that Algorithm 1 must converge to a stationary point and the algorithm cannot oscillate infinitely, that is, it terminates after a finite number of iterations.

Lemma 3: Given the fusion rule *F*0, for any initial values *I* ð Þ 0 <sup>1</sup> , … , *I* ð Þ 0 *L* � � in (26),

*CMC I* ð Þ *n* <sup>1</sup> , … , *I* ð Þ *n <sup>j</sup>* , *I* ð Þ *n <sup>j</sup>*þ1, … , *<sup>I</sup>* ð Þ *n <sup>L</sup>* ; *F*0; *Pce*0, *Pce*<sup>1</sup> , *N* � � must converge to a stationary point after a finite number of iterations.

**Proof:** For *j* ¼ 1, … , *L*, we denote *CMC* (20) in the *n* þ 1th iteration process by

$$\begin{split} & \mathbf{C}\_{\text{MC}} \Big( I\_{1}^{(n+1)}, \dots, I\_{j}^{(n+1)}, I\_{j+1}^{(n)}, \dots, I\_{L}^{(n)}; F^{\text{o}}; P^{\text{co}}, P^{\text{co}}, N \Big) \\ &= \frac{1}{N} \sum\_{i=1}^{N} \Big\{ \Big\{} \mathbbm{1} - I\_{j}^{(n+1)}(Y\_{ij}) \Big[ P\_{j1}(I\_{i}^{(n+1)}(Y\_{i1}), \dots, I\_{j-1}^{(n+1)}(Y\_{i(j-1)}), I\_{j+1}^{(n)}(Y\_{i(j+1)}), \dots, I\_{L}^{(n)}(Y\_{iL}); F^{\text{o}}; \\ & P^{\text{co}}, P^{\text{c1}}, N) + P\_{j2}(I\_{1}^{(n+1)}(Y\_{i1}), \dots, I\_{j-1}^{(n+1)}(Y\_{i(j-1)}), I\_{j+1}^{(n)}(Y\_{i(j+1)}), \dots, I\_{L}^{(n)}(Y\_{iL}); F^{\text{o}}; P^{\text{c1}}, P^{\text{c1}}, N) \Big\} \\ & \qquad \qquad \frac{\hat{L}(Y\_{i})}{\mathbf{g}(Y\_{i})} + c. \tag{31} \end{split} \tag{32}$$

Similarly, we denote the ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> th iteration process of the iterative items *Pj*1ð Þ� *<sup>L</sup>*^ð Þ� in (27)–(29) by

$$\mathbf{G}\_{j}^{i} = \text{Pr}\_{\text{I}}\Big(I\_{\text{1}}^{(n+1)}(Y\_{i\text{1}}), \dots, I\_{j-1}^{(n+1)}(Y\_{i(j-1)}), I\_{j+1}^{(n)}(Y\_{i(j+1)}), \dots, I\_{L}^{(n)}(Y\_{i\text{L}}); F^{0}; P^{x0}, P^{x1}, N\Big)\hat{L}(Y\_{i}), \quad \text{(32)}$$

for *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>* and *<sup>j</sup>* <sup>¼</sup> 1, … , *<sup>L</sup>:* Plugging *<sup>G</sup><sup>i</sup> <sup>j</sup>* into (31), we know

$$\mathbf{C\_{MC}}\left(I\_1^{(n+1)},\ldots,I\_j^{(n+1)},I\_{j+1}^{(n)},\ldots,I\_L^{(n)};F^0;P^{x0},P^{x1},N\right) = \frac{\mathbf{1}}{N} \sum\_{i=1}^N \frac{\left[1-I\_j^{(n+1)}(Y\_{ij})\right]}{\mathbf{g}(Y\_i)}\mathbf{G\_j^i+\mathbf{C}\_j^i},\tag{33}$$

where *C<sup>i</sup> <sup>j</sup>* <sup>¼</sup> *<sup>c</sup>* <sup>þ</sup> <sup>1</sup> *N* P*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup>*Pj*<sup>2</sup> *<sup>I</sup>* ð Þ *n*þ1 <sup>1</sup> ð Þ *Yi*<sup>1</sup> , … , *I* ð Þ *n*þ1 *<sup>j</sup>*�<sup>1</sup> *Yi j*ð Þ �<sup>1</sup> � �, *I* ð Þ *n <sup>j</sup>*þ<sup>1</sup> *Yi j*ð Þ <sup>þ</sup><sup>1</sup> � �, … , *I* ð Þ *n <sup>L</sup>* ð Þ *YiL* ; *<sup>F</sup>*0; � *Pce*0, *Pce*<sup>1</sup> , *<sup>N</sup>*<sup>Þ</sup> *L Y* ^ð Þ*<sup>i</sup> g Y*ð Þ*<sup>i</sup>* is independent of *<sup>I</sup>* ð Þ *n <sup>j</sup>* and *I* ð Þ *n*þ1 *<sup>j</sup>* . Splitting 1 � *I* ð Þ *n*þ1 *<sup>j</sup> Yij* � � into two terms, we obtain

$$\begin{split} & \quad \text{C\_{MC}} \left( I\_1^{(n+1)}, \dots, I\_j^{(n+1)}, I\_{j+1}^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\varepsilon 0}, P^{\varepsilon 1}, N \right) \\ & \quad = \frac{1}{N} \sum\_{i=1}^N \frac{\left[ 1 - I\_j^{(n)}(Y\_{\vec{y}}) \right] + \left[ I\_j^{(n)}(Y\_{\vec{y}}) - I\_j^{(n+1)}(Y\_{\vec{y}}) \right]}{g(Y\_i)} G\_j^i + G\_j^i \\ & \quad = \frac{1}{N} \sum\_{i=1}^N \frac{\left[ 1 - I\_j^{(n)}(Y\_{\vec{y}}) \right]}{g(Y\_i)} G\_j^i + G\_j^i + \frac{1}{N} \sum\_{i=1}^N \frac{\left[ I\_j^{(n)}(Y\_{\vec{y}}) - I\_j^{(n+1)}(Y\_{\vec{y}}) \right]}{g(Y\_i)} G\_j^i \\ & \quad = C\_{\text{MC}} \left( I\_1^{(n+1)}, \dots, I\_{j-1}^{(n+1)}, I\_j^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\varepsilon 0}, P^{\varepsilon 1}, N \right) + D\_j^{(n+1)}, \end{split} \tag{34}$$

where

$$D\_j^{(n+1)} = \frac{1}{N} \sum\_{i=1}^{N} \frac{\left[I\_j^{(n)}\left(\mathbf{Y}\_{\vec{\eta}}\right) - I\_j^{(n+1)}\left(\mathbf{Y}\_{\vec{\eta}}\right)\right]}{\mathbf{g}(\mathbf{Y}\_i)} \mathbf{G}\_j^i. \tag{35}$$

Note that (27)–(29) imply that *I* ð Þ *n*þ1 *<sup>j</sup> Yij* � � <sup>¼</sup> 0 if and only if *Gi <sup>j</sup>* < 0 and *I* ð Þ *n*þ1 *<sup>j</sup> Yij* � � <sup>¼</sup> 1 if and only if *<sup>G</sup><sup>i</sup> <sup>j</sup>* ≥0 for *i* ¼ 1, … , *N*, *j* ¼ 1, … , *L*. It means

$$\left[I\_j^{(n)}\left(\mathbf{Y}\_{i\bar{\jmath}}\right) - I\_j^{(n+1)}\left(\mathbf{Y}\_{i\bar{\jmath}}\right)\right] \mathbf{G}\_j^i \le \mathbf{0}.\tag{36}$$

Thus, for ∀*i*, *j*

$$\left[I\_j^{(n)}\left(Y\_{\vec{\eta}}\right) - I\_j^{(n+1)}\left(Y\_{\vec{\eta}}\right)\right] \mathbf{G}\_j^i / \mathbf{g}\left(Y\_i\right) \le \mathbf{0},\tag{37}$$

where the inequality (35) holds since *g*ð Þ� is well-defined (i.e., *g*ð Þ� >0). Substituting (35) into (33) yields *D*ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>j</sup>* ≤0. Thus, for ∀*j*≤*L*,

$$\begin{split} & \mathbf{C\_{MC}} \left( I\_1^{(n+1)}, \dots, I\_j^{(n+1)}, I\_{j+1}^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\varepsilon 0}, P^{\varepsilon 1}, N \right) \\ & \leq \mathbf{C\_{MC}} \left( I\_1^{(n+1)}, \dots, I\_{j-1}^{(n+1)}, I\_j^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\varepsilon 0}, P^{\varepsilon 1}, N \right). \end{split} \tag{38}$$

Furthermore,

$$\begin{split} & \mathbf{C\_{MC}} \left( I\_1^{(n+1)}, I\_2^{(n+1)}, \dots, I\_L^{(n+1)}; F^0; P^{\epsilon 0}, P^{\epsilon 1}, N \right) \\ & \leq \mathbf{C\_{MC}} \left( I\_1^{(n)}, I\_2^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\epsilon 0}, P^{\epsilon 1}, N \right) . \end{split} \tag{39}$$

It means *CMC* is nonincreasing. Note that *CMC I* ð Þ *n* <sup>1</sup> , *I* ð Þ *n* <sup>2</sup> , … , *I* ð Þ *n <sup>L</sup>* ; *F*0; *Pce*0, *Pce*<sup>1</sup> , *N* � � is a finite value. We conclude that it must converge to a stationary point after a finite number of iterations.

Theorem 1.2: Given the fusion rule *F*0, the sensor rules *I* ð Þ *n* <sup>1</sup> , *I* ð Þ *n* <sup>2</sup> , … , *I* ð Þ *n <sup>L</sup>* are finitely convergent, i.e., Algorithm 1 converges after a finite number of iterations.

**Proof:** By Lemma 3, *CMC* must attain a stationary point after a finite number of iterations. It means that the value of *CMC* cannot change after *n*th iteration, that is,

$$\begin{aligned} \mathbf{C\_{MC}} \left( I\_1^{(n+1)}, \dots, I\_j^{(n+1)}, I\_{j+1}^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\epsilon 0}, P^{\epsilon 1}, N \right) \\ = \mathbf{C\_{MC}} \left( I\_1^{(n+1)}, \dots, I\_{j-1}^{(n+1)}, I\_j^{(n)}, \dots, I\_L^{(n)}; F^0; P^{\epsilon 0}, P^{\epsilon 1}, N \right). \end{aligned} \tag{40}$$

Using (32) and (37), we derive that *D*ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>j</sup>* ¼ 0. Combining (33)–(35), we know

$$\left[I\_j^{(n)}\left(Y\_{\vec{\eta}}\right) - I\_j^{(n+1)}\left(Y\_{\vec{\eta}}\right)\right] \mathbf{G}\_j^i = \mathbf{0}, for \, i = 1, \ldots, N,\tag{41}$$

which implies either *I* ð Þ *n <sup>j</sup> Yij* � � � *<sup>I</sup>* ð Þ *n*þ1 *<sup>j</sup> Yij* � � <sup>¼</sup> 0, *<sup>i</sup>:e:*, *<sup>I</sup>* ð Þ *n <sup>j</sup> Yij* � � <sup>¼</sup> *<sup>I</sup>* ð Þ *n*þ1 *<sup>j</sup> Yij* � � or *<sup>G</sup><sup>i</sup> <sup>j</sup>* ¼ 0, *i:e:*, *I* ð Þ *n*þ1 *<sup>j</sup> Yij* � � <sup>¼</sup> 1, *<sup>I</sup>* ð Þ *n <sup>j</sup> Yij* � � <sup>¼</sup> <sup>0</sup>*:* It follows that when *CMC* converges to a stationary point, either *I* ð Þ *n*þ1 *<sup>j</sup> Yij* � � is invariant or *<sup>I</sup>* ð Þ *n*þ1 *<sup>j</sup> Yij* � � <sup>¼</sup> 1, *<sup>I</sup>* ð Þ *n <sup>j</sup> Yij* � � <sup>¼</sup> 0. Namely, *<sup>I</sup>* ð Þ *n*þ1 *<sup>j</sup> Yij* � � can only change from 0 to 1 at most a finite number of times. Therefore, the *I* ð Þ *n* <sup>1</sup> , *I* ð Þ *n* <sup>2</sup> , … , *I* ð Þ *n <sup>L</sup>* are finitely convergent.
