## 3 Main Results
\right] \tag{47}\]  
_where_  
\[\hat{\mathbf{x}}_{k|k}=\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{k}^{x}\Big{(}\mathbf{y}_{k}^{*}- \hat{\mathbf{y}}_{k|k-1}^{*}\Big{)} \tag{48}\]  
\[\mathbf{P}_{k|k}=\frac{(v_{3}-2)\Big{(}v_{3}+\Big{(}\Delta_{k}^{*}\Big{)}^{2} \Big{)}}{v_{3}(v_{3}+m-2)}\Big{(}\mathbf{P}_{k|k-1}-\mathbf{K}_{k}\mathbf{P}_{k|k-1}^{yy} \mathbf{K}_{k}^{\mathsf{T}}\Big{)} \tag{49}\]  
\[\mathbf{K}_{k}^{x}=\mathbf{P}_{k|k-1}^{xy}\Big{(}\mathbf{P}_{k|k-1}^{yy}\Big{)}^{-1} \tag{50}\]  
\[\mathbf{P}_{k|k-1}^{xy}=(1-p_{k})\mathbf{P}_{k|k-1}^{xz}+p_{k}\mathbf{P}_{k-1|k-1}^{xz} \tag{51}\]  
\[\mathbf{P}_{k|k}^{\eta\eta}=-\mathbf{K}_{k}^{x}\mathbf{P}_{k|k-1}^{yy}\Big{(}\mathbf{K}_{k}^{ \prime}\Big{)}^{\mathsf{T}} \tag{52}\]  
_where \(\mathbf{K}_{k}^{x}\) is the filtering gain of states, and_  
\[\hat{\mathbf{x}}_{k|k-1}=\int_{\mathbb{R}^{a}}\mathbf{A}_{k-1}\mathbf{x}_{k-1}\mathsf{St} \Big{(}\mathbf{x}_{k-1};\hat{\mathbf{x}}_{k-1|k-1},\mathbf{P}_{k-1|k-1},v_{3}\Big{)}\text{d} \mathbf{x}_{k-1} \tag{53}\]  
\[\mathbf{P}_{k|k-1}^{xz}=\frac{v_{3}-2}{v_{3}}\int_{\mathbb{R}^{a}}\mathbf{A}_{k-1}\mathbf{x }_{k-1}\mathbf{x}_{k-1}^{\mathsf{T}}\mathbf{A}_{k-1}^{\mathsf{T}}\mathbf{A}_{k-1}^{\mathsf{T }}\mathsf{St}\Big{(}\mathbf{x}_{k-1};\hat{\mathbf{x}}_{k-1|k-1},\mathbf{P}_{k-1|k-1},v_{3} \Big{)}\text{d}\mathbf{x}_{k-1} \tag{54}\]  
\[-\frac{v_{3}-2}{v_{3}}\hat{\mathbf{x}}_{k|k-1}\mathbf{x}_{k-1}^{\mathsf{T}}+\frac{v_{1 }(v_{3}-2)}{(v_{1}-2)v_{3}}\mathbf{Q}_{k-1} \tag{55}\]  
\[\mathbf{P}_{k|k-1}^{xz}=\frac{v_{3}-2}{v_{3}}\bigg{(}\int x_{\mathbf{x}}(\mathbf{x}_{k})^{ \mathsf{T}}\mathbf{F}_{k}^{\mathsf{T}}\mathsf{St}\Big{(}\mathbf{x}_{k};\hat{\mathbf{x}}_{k|k -1},\mathbf{P}_{k|k-1},v_{3}\Big{)}\text{d}\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{z} }_{k|k-1}^{*}+\mathbf{S}_{k}\Big{)} \tag{56}\]  
\[\mathbf{P}_{k|k-1}^{xz}=\frac{v_{3}-2}{v_{3}}\bigg{(}\int x_{\mathbf{x}}(\mathbf{x}_{k})^{ \mathsf{T}}\mathbf{F}_{k}^{\mathsf{T}}\mathsf{St}\Big{(}\mathbf{x}_{k};\hat{\mathbf{x}}_{k|k -1},\mathbf{P}_{k|k-1},v_{3}\Big{)}\text{d}\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{z} }_{k|k-1}^{*}+\mathbf{S}_{k}\Big{)} \tag{57}\]  
**Proof. ** First, \(\hat{x}_{k|k}\), \(\mathbf{P}_{k|k}\), \(\mathbf{K}_{k}^{x}\), \(\hat{x}_{k|k-1}\), and \(\mathbf{P}_{k|k-1}\) can be obtained by definition. Then, substituting Equation (28) to the definition of \(\mathbf{P}_{k|k-1}^{xy}=E\Big{[}\overline{x}_{k|k-1}\overline{y}_{k|k-1}^{*}|\mathbf{ Y}_{k-1}^{*}\Big{]}\), we can easily obtain Equation (51). Second, according to the definition, \(\mathbf{P}_{k|k-1}^{xz}\) is given as:  
\[ \begin{array}{ll}\mathbf{P}_{k|k-1}^{xz}&=E\Big{[}\Big{(}\mathbf{x}_{k}-\hat{x}_{k|k- 1}\Big{)}\Big{(}\mathbf{z}_{k}^{*}-\hat{z}_{k|k-1}^{*}\Big{)}^{\mathrm{T}}|\mathbf{Y}_{ k-1}^{*}\Big{]}\\ &=E\Big{[}\mathbf{x}_{k}(\mathbf{x}_{k})^{\mathrm{T}}\mathbf{F}_{k}^{\mathrm{T}}|\mathbf{Y}_{ k-1}^{*}\Big{]}-\frac{y_{2}-2}{v_{3}}\hat{x}_{k|k-1}\hat{z}_{k|k-1}^{*}+E\Big{[} \mathbf{\omega}_{k-1}\mathbf{\eta}_{k}^{\mathrm{T}}|\mathbf{Y}_{k-1}^{*}\Big{]}\end{array}  \tag{57}\]  
Then, according to the definition, \(\mathbf{P}_{k-1|k-1}^{xz}\) can be obtained as:  
\[ \begin{array}{ll}\mathbf{P}_{k-1|k-1}^{xz}&=E\Big{[}\Big{(}\mathbf{x}_{k}-\hat{x}_{ k|k-1}\Big{)}\Big{(}\mathbf{z}_{k-1}^{*}-\hat{z}_{k|k-1}^{*}\Big{)}^{\mathrm{T}}|\mathbf{Y}_{ k-1}^{*}\Big{]}\\ &=E\Big{[}\mathbf{A}_{k-1}\mathbf{x}_{k-1}(\mathbf{F}_{k-1}\mathbf{x}_{k-1}+\mathbf{\eta}_{k-1})^{ \mathrm{T}}|\mathbf{Y}_{k-1}^{*}\Big{]}-\frac{v_{4}-2}{v_{4}}\hat{x}_{k|k-1}\Big{(} \hat{z}_{k-1|k-1}^{*}\Big{)}^{\mathrm{T}}\end{array}  \tag{58}\]  
Under the Student's t approximation of \(p\Big{(}\mathbf{x}_{k-1}^{a}|\mathbf{Y}_{k-1}^{*}\Big{)}\) with the known \(\hat{x}_{k-1|k-1}^{a}\) and \(\mathbf{P}_{k-1|k-1}^{a}\), we can get Equation (56). Finally, using Equations (30) and (48), we can get  
\[ \begin{array}{ll}\overline{\eta}_{k|k}=\eta_{k}-\mathbf{K}_{k}^{\mathrm{T}} \overline{y}_{k|k-1}^{*}\\ \overline{x}_{k|k}=\overline{x}_{k|k-1}-\mathbf{K}_{k}^{X\overline{y}} _{k|k-1}\end{array}