# 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

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

• You might find my account of the bivariate case at stats.stackexchange.com/a/71303/919 to be helpful, especially at the remark "the vertical skew transformation rescales each conditional distribution by $\sqrt{1-\rho^2}$ and then recenters it by $\rho x.$"
– whuber
Commented Aug 2, 2021 at 0:52
• You write down the joint density and then treat $x_1$ as a constant. What do you obtain? Commented Aug 2, 2021 at 6:44

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