## 2 Preliminaries and Problem Statement
### Problem Statement
\begin{array}{cc}\hat{\mathbf{x}}_{k|k}\\ \hat{\mathbf{x}}_{k+1|k}\end{array} \right]\Bigg{[} \begin{array}{cc}\mathbf{P}_{k|k}& \mathbf{P}_{k|k+1|k}\\ \mathbf{P}_{kk+1|k}^{\mathrm{T}}&\mathbf{P}_{k+1|k}\end{array} \right]v_{3}\Bigg{)}. \tag{12}\]  
The framework is shown:  
1. Time prediction  
\[\hat{\mathbf{x}}_{k|k-1}=\int_{\mathbb{R}^{n}}\!\!A_{k-1}x_{k-1}\mathrm{St}\Big{(} \mathbf{x}_{k-1};\hat{\mathbf{x}}_{k-1|k-1},\mathbf{P}_{k-1|k-1},v_{3}\Big{)}\mathrm{d}x_{k-1} \tag{13}\]  
\[ \begin{array}{ll}\mathbf{P}_{k|k-1}=&\frac{v_{3}-2}{v_{3}}\int_{\mathbb{R}^{n}} \mathbf{A}_{k-1}x_{k-1}x_{k-1}^{\mathrm{T}}\mathrm{St}\Big{(}\mathbf{x}_{k-1};\hat\mathbf{{x}}_{k-1|k-1},\mathbf{P}_{k-1|k-1},v_{3}\Big{)}\mathrm{d}\mathbf{x}_{k-1}\\ &-\frac{v_{3}-2}{v_{3}}\hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{x}}_{k|k-1}^{\mathrm{T}}+ \frac{v_{1}(v_{3}-2)}{(v_{1}-2)v_{3}}\mathbf{Q}_{k-1}\end{array}  \tag{14}\]  
where \((\bullet)^{\mathrm{T}}\) is the transpose operation, \(\hat{\mathbf{x}}_{k|k-1}\) and \(\mathbf{P}_{k|k-1}\) are the one-step predicted vector and the corresponding covariance matrix, respectively. 2. Measurement update  
\[\Delta_{k}=\sqrt{\Big{(}\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\Big{)}^{\mathrm{T}} \Big{(}\mathbf{P}_{k|k-1}^{zz}\Big{)}^{-1}\Big{(}\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1} \Big{)}} \tag{15}\]  
\[ \begin{array}{c}\mathbf{K}_{k}=\mathbf{P}_{k|k-1}^{xz}\Big{(}\mathbf{P}_{k|k-1}^{xz} \Big{)}^{-1}\end{array}  \tag{16}\]  
\[\hat{\mathbf{x}}_{k|k}=\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{k}\Big{(}\mathbf{z}_{k}-\hat{\mathbf{z}}_ {k|k-1}\Big{)} \tag{17}\]\[\mathbf{P}_{k\!k}=\frac{(v_{3}-2)\big{(}v_{3}+\Delta_{k}^{2}\big{)}}{v_{3}(v_{3}+m-2)} \Big{(}\mathbf{P}_{k\!|k-1}-\mathbf{K}_{k}\mathbf{P}_{k\!|k-1}^{zz}\mathbf{K}_{k}^{\mathrm{T}} \Big{)} \tag{18}\]  
where \((\bullet)^{-1}\) is the inverse operation, \(\hat{\mathbf{x}}_{k\!|k}\) and \(\mathbf{P}_{k\!|k}\) are the estimating state and the corresponding covariance matrix, respectively, \(\hat{\mathbf{z}}_{k\!|k-1}\) and \(\mathbf{P}_{k\!|k-1}^{zz}\) are the predicting measurement and its covariance matrix, respectively, \(\mathbf{P}_{k\!|k-1}^{zz}\) is the cross-covariance matrix of the state and measurement vectors. The formulas of them are as follows:  
\[\hat{\mathbf{z}}_{k\!|k-1}=\int_{\mathbb{R}^{n}}\mathbf{H}_{k}\mathbf{x}_{k-1}\mathrm{St} \Big{(}\mathbf{x}_{k};\hat{\mathbf{x}}_{k\!|k-1},\mathbf{P}_{k\!|k-1},v_{3}\Big{)}\mathrm{ d}\mathbf{x}_{k} \tag{19}\]  
\[\mathbf{P}_{k\!|k-1}^{zz}= \frac{v_{3}-2}{v_{3}}\int_{\mathbb{R}^{n}}\mathbf{H}_{k}\mathbf{x}_{k} \mathbf{x}_{k}^{\mathrm{T}}\mathbf{H}_{k}^{\mathrm{T}}\mathrm{St}\Big{(}\mathbf{x}_{k}; \hat{\mathbf{x}}_{k\!|k-1},\mathbf{P}_{k\!|k-1},v_{3}\Big{)}\mathrm{d}\mathbf{x}_{k} \tag{20}\] \[-\frac{v_{3}-2}{v_{3}}\sum_{k\!|k-1}\mathbf{\pi}_{kk-1}^{\mathrm{T}}+ \frac{v_{2}(v_{3}-2)}{(v_{2}-2)v_{3}}\mathbf{R}_{k}\] \[\mathbf{P}_{k\!|k-1}^{zz}= \frac{v_{3}-2}{v_{3}}\sum_{\mathbb{R}^{n}}\mathbf{x}_{k}\mathbf{x}_{k}^{ \mathrm{T}}\mathbf{H}_{k}^{\mathrm{T}}\times\mathrm{St}\Big{(}\mathbf{x}_{k};\hat{\mathbf{ x}}_{k\!|k-1},\mathbf{P}_{k\!|k-1},v_{3}\Big{)}\mathrm{d}\mathbf{x}_{k}\] (21) \[-\frac{v_{3}-2}{v_{3}}\sum_{k\!|k-1}\mathbf{\pi}_{k\!|k-1}^{\mathrm{T}}\]  
In practice, the time-correlation of the measurement noises often originates from a time-correlated noisy work environment or the dependence on the states. Otherwise, the STD is suitable to describe the heavy-tailed noises. The goal of the paper was to design an estimator for systems with Student's t distributional time-correlated noises.