**Abbreviations**

The following abbreviations are used in this manuscript:



#### **Appendix A. Power Series Expansion of Cauchy Kernel Function**

The binomial expansion of (1 + *x*)−*<sup>N</sup>* for negative integer − *N* is given as follows:

$$\begin{aligned} \left(1+\mathbf{x}\right)^{-N} &= 1 + (-N)\mathbf{x} + \frac{(-N)(-N-1)}{2!} \mathbf{x}^2 + \frac{(-N)(-N-1)(-N-2)}{3!} \mathbf{x}^3 + \dotsb \\ &= \sum\_{k=0}^{\infty} (-1)^k \binom{N+k-1}{k} \mathbf{x}^k \quad \text{for } |\mathbf{x}| < 1. \end{aligned}$$

Now the correntropy measure, by taking the binomial series expansion of Cauchy kernel with *x* = (*<sup>X</sup>*−*<sup>Y</sup>*)<sup>2</sup> *δ*is

$$\begin{split} V\_{\delta}(X,Y) &= \sum\_{k=0}^{\infty} (-1)^{k} \binom{N+k-1}{k} \left( \frac{(X-Y)^{2}}{\delta} \right)^{k} \\ &= \sum\_{k=0}^{\infty} \frac{(-1)^{k}}{\delta^{k}} \binom{N+k-1}{k} E\left[ (X-Y)^{2k} \right], \quad \text{for} \left| \frac{(X-Y)^{2}}{\delta} \right| < 1. \end{split}$$

#### **Appendix B. Derivation of Kalman Gain**

The Kalman gain for Gaussian kernel based nonlinear estimator is **K***<sup>G</sup> k* with **L***<sup>G</sup> k* as a scalar term. Similarly, for Cauchy kernel, it is **K***<sup>C</sup> k* and **L***<sup>C</sup> k* respectively. A general expression for Kalman gain is given as **K***k* = (**<sup>P</sup>**−<sup>1</sup> *k*|*k*−<sup>1</sup> + **<sup>L</sup>***k***H R**−<sup>1</sup> *k* **<sup>H</sup>**)−1**L***k***H R**−<sup>1</sup> *k* . Applying matrix inversion lemma

$$\begin{split} \mathbf{K}\_{k} &= \left(\mathbf{P}\_{k|k-1} - \mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'(\mathbf{R}\_{k} + \overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}')^{-1}\overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\right)\mathbf{L}\_{k}\overline{\mathbf{H}}'\mathbf{R}\_{k}^{-1} \\ &= \mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\mathbf{R}\_{k}^{-1} - \mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'(\mathbf{R}\_{k} + \overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}')^{-1}\overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\mathbf{R}\_{k}^{-1} \\ &= \mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\left(\mathbf{R}\_{k}^{-1} - (\mathbf{R}\_{k} + \overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}')^{-1}\overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\mathbf{R}\_{k}^{-1}\right) \\ &= \mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\left(\mathbf{I} - (\mathbf{R}\_{k} + \overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}')^{-1}\overline{\mathbf{H}}\mathbf{P}\_{k|k-1}\mathbf{L}\_{k}\overline{\mathbf{H}}'\right)\mathbf{R}\_{k}^{-1}. \end{split}$$

After certain algebraic manipulations, we ge<sup>t</sup>

$$\mathbf{K}\_{k} = \mathbf{P}\_{k|k-1} \mathbf{L}\_{k} \overline{\mathbf{H}}^{\prime} (\mathbf{I} + \mathbf{R}\_{k}^{-1} \overline{\mathbf{H}} \mathbf{P}\_{k|k-1} \mathbf{L}\_{k} \overline{\mathbf{H}}^{\prime})^{-1} \mathbf{R}\_{k}^{-1} = \mathbf{P}\_{k|k-1} \mathbf{L}\_{k} \overline{\mathbf{H}}\_{k}^{\prime} (\mathbf{R}\_{k} + \overline{\mathbf{H}}\_{k} \mathbf{P}\_{k|k-1} \mathbf{L}\_{k} \overline{\mathbf{H}}\_{k}^{\prime})^{-1}. \tag{A1}$$

From the above equation, **K***<sup>G</sup> k* and **K***<sup>C</sup> k* can be defined by making necessary substitution for **L***<sup>G</sup> k* and **L***<sup>C</sup> k* .
