## 2 Preliminaries and Problem Statement  
### Problem Statement  
Consider the following classic discrete stochastic system with time-correlated measurement noise:  
\[\mathbf{x}_{k+1}=\mathbf{A}_{k}\mathbf{x}_{k}+\mathbf{\omega}_{k} \tag{1}\]  
\[\mathbf{z}_{k}=\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{\xi}_{k} \tag{2}\]  
\[\mathbf{\xi}_{k+1}=\mathbf{C}_{k}\mathbf{\xi}_{k}+\mathbf{\epsilon}_{k} \tag{3}\]  
where \(\mathbf{x}_{k}\in\mathbb{R}^{n}\) is the state vector; \(\mathbf{z}_{k}\in\mathbb{R}^{m}\) is the measurement vector; \(\mathbf{A}_{k}\in\mathbb{R}^{m\times n}\), \(\mathbf{H}_{k}\in\mathbb{R}^{m\times n}\), and \(\mathbf{C}_{k}\in\mathbb{R}^{m\times m}\) are known transition matrices; \(\mathbf{\omega}_{k}\in\mathbb{R}^{n}\), and \(\mathbf{\xi}_{k}\in\mathbb{R}^{m}\) are the heavy-tailed process noise vector, the time-correlated heavy-tailed measurement noise vector respectively. The heavy-tailed noises are caused by system outliers, which are modeled as STD:  
\[p(\mathbf{\omega}_{k})=\text{St}(\mathbf{\omega}_{k};0,\mathbf{Q}_{k},v_{1}) \tag{4}\]  
\[p(\mathbf{\epsilon}_{k})=\text{St}(\mathbf{\epsilon}_{k};0,R_{k},v_{2}) \tag{5}\]  
where \(\text{St}(x;\mu,\Sigma,v)\) is the Student's t probability density function (PDF); \(\mu\) is the mean value; \(\Sigma\) is the scale matrix; \(v\) is the dof parameter. The initial state \(\mathbf{x}_{0}\) is also assumed to satisfy STD, and the PDF is as follows:  
\[p(\mathbf{x}_{0})=\text{St}\Big{(}\mathbf{x}_{0};\hat{\mathbf{x}}_{0|0},P_{0|0},v_{3} \Big{)}. \tag{6}\]  
Furthermore, \(\mathbf{x}_{0}\), \(\mathbf{\omega}_{k}\) and \(\mathbf{\xi}_{k}\) are assumed to be mutually independent. According to Reference [13], for STD, the conditional estimating vector \(\hat{\mathbf{x}}_{k|N}\) and the corresponding covariance matrix \(P_{k|N}\) can be approximated as:  
\[\hat{\mathbf{x}}_{k|N}=E[\mathbf{x}_{k}|\mathbf{Z}_{N}]=\int\mathbf{x}_{k}p(\mathbf{x}_{k}|\mathbf{Z}_ {N})\text{d}\mathbf{x}_{k} \tag{7}\]\[\mathbf{P}_{k|N}=\frac{v_{3}-2}{v_{3}}E\Big{[}\widetilde{\mathbf{x}}_{k|N}\widetilde{\mathbf{x }}_{k|N}^{\mathrm{T}}|\mathbf{Z}_{N}\Big{]}=\frac{v_{3}-2}{v_{3}}\int\widetilde\mathbf{{x}}_{k|N}\widetilde{\mathbf{x}}_{k|N}^{\mathrm{T}}p(x_{k}|\mathbf{Z}_{N})\mathrm{d}\mathbf{ x}_{k} \tag{8}\]  
where \(\widetilde{\mathbf{x}}_{k|N}=\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|N}\) denotes the estimating error. The one-step delayed measurement is described as:  
\[\mathbf{y}_{k}=(1-\gamma_{k})\mathbf{z}_{k}+\gamma_{k}\mathbf{z}_{k-1},\ k>1;\ y_{1}=\mathbf{z }_{1} \tag{9}\]  
where \(\gamma_{k}\) is a sequence of Bernoulli random variables with a value of 0 or 1, i.e.,  
\[ \begin{array}{l}p(\gamma_{k}=1)=E[\gamma_{k}]=p_{k},\\ p(\gamma_{k}=0)=1-E[\gamma_{k}]=1-p_{k},\\ E\Big{[}(\gamma_{k}-p_{k})^{2}\Big{]}=(1-p_{k})p_{k}\end{array}  \tag{10}\]  
Here, \(p_{k}\) is the probability of delay occurrence at time \(k\). Remark 1: _Referring to Reference [13], for a vector \(x\) with PDF \(p(x)=\mathrm{St}(x;\mu,\Sigma,v)\), the corresponding covariance matrix satisfies \(\frac{v-2}{v}\Sigma\), i.e., \(E\Big{[}(x-\mu)(x-\mu)^{\mathrm{T}}\Big{]}=\frac{v-2}{v}\Sigma\)._  
### Problem Statement  
In Reference [13], a Student's t based filter framework is designed. Firstly, there are two assumptions as follows:  
**Assumption 1**. : _The joint PDF \(p(\mathbf{x}_{k},\mathbf{z}_{k}|\mathbf{Z}_{k-1})\) satisfies STD, i.e.,_  
\[p(\mathbf{x}_{k},\mathbf{z}_{k}|\mathbf{Z}_{k-1})=\mathrm{St}\Bigg{(}\left[ \begin{array}[] {c}\mathbf{x}_{k}\\ \mathbf{z}_{k}\end{array} \right];\left[ \begin{array}{c}\hat{\mathbf{x}}_{k|k-1}\\ \hat{\mathbf{z}}_{k|k-1}\end{array} \right]\Bigg{[} \begin{array}{cc}\mathbf{P}_{k|k- 1}&\mathbf{P}_{k|k-1}^{xz}\\ \mathbf{P}_{k|k-1}^{xz}\end{array} \right]v_{3}\Bigg{)}. \tag{11}\]  
**Assumption 2**. : _The joint PDF \(p(\mathbf{x}_{k},\mathbf{x}_{k+1}|\mathbf{Z}_{k})\) satisfies STD, i.e.,_  
\[p(\mathbf{x}_{k},\mathbf{x}_{k+1}|\mathbf{Z}_{k})=\mathrm{St}\Bigg{(}\left[ \begin{array}[] {c}\mathbf{x}_{k}\\ \mathbf{x}_{k+1}\end{array} \right];\left[