## 3 Main Results  
The measurement noise is no longer time irrelevant. To solve the problem, according to Reference [24], the measurement differencing method was used. We defined the equivalent measurement \(\mathbf{z}_{k}^{\star}=\mathbf{z}_{k}-\mathbf{C}_{k-1}\mathbf{z}_{k-1}\) and Equations (1)-(3) can be converted to the following form:  
\[\mathbf{x}_{k+1}=\mathbf{A}_{k}\mathbf{x}_{k}+\mathbf{\omega}_{k} \tag{22}\]  
\[\mathbf{z}_{k}^{\star}=\mathbf{F}_{k}\mathbf{x}_{k}+\mathbf{\eta}_{k} \tag{23}\]  
where \(\mathbf{F}_{k}=\mathbf{H}_{k}-\mathbf{\Gamma}_{k-1}\), \(\mathbf{\eta}_{k}=\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}+\mathbf{\epsilon}_{k-1}\), and \(\mathbf{\Gamma}_{k-1}=\mathbf{C}_{k-1}\mathbf{H}_{k-1}\mathbf{A}_{k-1}^{-1}\). After transformation, \(\mathbf{\xi}_{k}\) has been rewritten as an equivalent measurement noise \(\mathbf{\eta}_{k}\), which consists of Student's t distributional noises \(\mathbf{\omega}_{k-1}\) and \(\mathbf{\epsilon}_{k-1}\). **Theorem 1**: _Denoting that \(\mathbf{\Xi}_{k}\) is the covariance of \(\mathbf{\eta}_{k}\), and \(\mathbf{S}_{k}\) is the cross-covariance of \(\mathbf{\omega}_{k-1}\) and \(\mathbf{\eta}_{k}\), we can derive:_  
\[\mathbf{\Xi}_{k}=\frac{v_{1}}{v_{1}-2}\mathbf{\Gamma}_{k-1}\mathbf{Q}_{k-1}\mathbf{\Gamma}_{k- 1}^{\mathrm{T}}+\frac{v_{2}}{v_{2}-2}\mathbf{R}_{k-1} \tag{24}\]  
\[\mathbf{S}_{k}=\frac{v_{1}}{v_{1}-2}\mathbf{Q}_{k-1}\mathbf{\Gamma}_{k-1}^{\mathrm{T}}. \tag{25}\]  
**Proof. ** Firstly, the expectation of \(\mathbf{\eta}_{k}\) is \(E[\mathbf{\eta}_{k}]=E[\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}+\mathbf{\epsilon}_{k-1}]\). As \(\mathbf{\omega}_{k}\) and \(\mathbf{\xi}_{k}\) are independent, \(E[\mathbf{\eta}_{k}]=E[\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}]+E[\mathbf{\epsilon}_{k-1}]=0\). Then, we have  
\[ \begin{array}{ll}\Xi_{k}&=E\Big{[}\overline{\mathbf{\eta}}_{k}\overline{\mathbf{\eta} }_{k}^{\mathrm{T}}\Big{]}=E\Big{[}(\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}+\mathbf{ \epsilon}_{k-1})(\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}+\mathbf{\epsilon}_{k-1})^{ \mathrm{T}}\Big{]}\\ &=E\Big{[}\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k-1}\mathbf{\omega}_{k-1}^{\mathrm{T}}\mathbf{ \Gamma}_{k-1}^{\mathrm{T}}\Big{]}+E\Big{[}\mathbf{\epsilon}_{k-1}\mathbf{\epsilon}_{k-1 }^{\mathrm{T}}\Big{]}\end{array}  \tag{26}\]  
According to Remark 1, we can derive Equation (24). Likewise, the cross-covariance \(\mathbf{S}_{k}\) in Equation (25) can be derived as follows:  
\[\mathbf{S}_{k}=E\Big{[}\overline{\mathbf{\omega}}_{k-1}\overline{\mathbf{\eta}}_{k}^{ \mathrm{T}}\Big{]}=E\Big{[}\mathbf{\omega}_{k-1}(\mathbf{\Gamma}_{k-1}\mathbf{\omega}_{k- 1}+\mathbf{\epsilon}_{k-1})^{\mathrm{T}}\Big{]}=E\Big{[}\mathbf{\omega}_{k-1}\mathbf{\omega} _{k-1}^{\mathrm{T}}\mathbf{\Gamma}_{k-1}^{\mathrm{T}}\Big{]}+E\Big{[}\mathbf{\omega}_{k-1} \mathbf{\epsilon}_{k-1}^{\mathrm{T}}\Big{]}. \tag{27}\]Moreover, Equation (9) can be transformed into  
\[\mathbf{y}_{k}^{*}=(1-\gamma_{k})(\mathbf{F}_{k}\mathbf{x}_{k}+\mathbf{\eta}_{k})+\gamma_{k}(\mathbf{ F}_{k-1}\mathbf{x}_{k-1}+\mathbf{\eta}_{k-1}). \tag{28}\]  
From Equation (28), we find that the computation of \(E\Big{(}\mathbf{y}_{k}^{*}\mathbf{y}_{k}^{*}|\mathbf{Y}_{k-1}^{*}\Big{)}\) also depends on the computation of \(E\Big{(}\mathbf{\eta}_{k-1}\mathbf{\eta}_{k-1}^{*}|\mathbf{Y}_{k-1}^{*}\Big{)}\). Therefore, according to Reference [21], we need to augment the state vector as follows:  
\[\mathbf{x}_{k}^{a}=\Big{[} \begin{array}{cc}\mathbf{x}_{k}^{\mathsf{T}}&\mathbf{\eta}_{k}^ {\mathsf{T}}\end{array} \Big{]}^{\mathsf{T}}. \tag{29}\]  
\(\Box\)  
**Theorem 2**.