Conditional distribution of multivariate cauchy distribution

Question

In the example of multivariate normal distribution,

$$ \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix}\right) $$

Then it is known that

$$ \mathbf{x}_2 \mid \mathbf{x}_1 \sim \mathcal{N}(\mathbf{m}, \mathbf{S}) $$

where

$$ \mathbf{m} = \mu_2 + \Sigma_{21} \Sigma_{11}^{-1} (\mathbf{x}_1 - \mu_1) $$

$$ \mathbf{S} = \Sigma_{22} - \Sigma_{21} \Sigma_{11}^{-1} \Sigma_{12} $$

Next, how can I express the conditional distribution $\mathbf{x}_2 \mid \mathbf{x}_1$ when they are multivariate cauchy distributed random variables?

$$ \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \sim \mathcal{CAUCHY}\left(\begin{bmatrix} l_1 \\ l_2 \end{bmatrix}, \begin{bmatrix} \Gamma_{11} & \Gamma_{12} \\ \Gamma_{21} & \Gamma_{22} \end{bmatrix}\right) $$

If it can't be expressed analytically, I also want to know how about in

$$ \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \sim \mathcal{CAUCHY}\left(\begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix}, \begin{bmatrix} \Gamma_{11} & \Gamma_{12} \\ \Gamma_{21} & \Gamma_{22} \end{bmatrix}\right) $$

Answer 1

I found that there is a formula of conditional multivariate t distribution.

https://arxiv.org/pdf/1402.4306.pdf https://sigfin.org/?plugin=attach&refer=021-18&openfile=SIG-FIN-021-18.pdf

Then we can get the formula of conditional multivariate cauchy distribution by setting $\nu = 1$.

In conclusion, the conditional distribution of multivariate cauchy distribution may be a multivariate t distribution.