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

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.
