Derive the joint probability density function of differences of Gaussian Mixtures

Question

Consider a 3-variate random vector $(\epsilon_0, \epsilon_1, \epsilon_2)$ which is distributed as a Gaussian mixture: (with some abuse of notation) $$ f(\epsilon_0, \epsilon_1, \epsilon_2)=\underbrace{w_a \mathcal{N}\Big(\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, \begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}\Big)}_{\text{component (b)}} $$ where $f$ denotes the PDF, $(w_a, w_b)$ are the mixture weights, $\mathcal{N}(\mu, \Sigma)$ stays for 3-variate normal distribution with mean $\mu$ and var-cov matrix $\Sigma$.

Is it possible to derive $f(\epsilon_1-\epsilon_0, \epsilon_1-\epsilon_2)$? If yes, could you walk me through the derivation?

I'm confused on how to proceed:

• Deriving the (marginal) PDF of $(\epsilon_1-\epsilon_0)$ and $(\epsilon_1-\epsilon_2)$ when $\{\epsilon_0, \epsilon_1, \epsilon_2\}$ are mutually independent is explained here for example.

• Here, however, I don't have mutual independence. Moreover, I want the joint of pairs of differences.

Any hint would be extremely appreciated.

Answer 1

Let $\left (\begin{matrix} \epsilon_1-\epsilon_0 \\ \epsilon_1-\epsilon_2\end{matrix}\right) = \left (\begin{matrix} 1 & -1 & 0 \\ 1& 0 & -1 \end{matrix}\right)\left (\begin{matrix} \epsilon_0 \\ \epsilon_1\\\epsilon_2\end{matrix}\right) = A\epsilon$

Then $$f(\epsilon_0-\epsilon_1, \epsilon_0-\epsilon_2)=\underbrace{w_a \mathcal{N}\Big(A\begin{pmatrix} \mu_a\\ \mu_a\\ \mu_a \end{pmatrix}, A\begin{pmatrix} \sigma^2_a & \rho_a & \rho_a\\ \rho_a &\sigma^2_a & \rho_a\\ \rho_a & \rho_a &\sigma^2_a \\ \end{pmatrix}A'\Big)}_{\text{component (a)}} +\underbrace{w_b\mathcal{N}\Big(A\begin{pmatrix} \mu_b\\ \mu_b\\ \mu_b \end{pmatrix}, A\begin{pmatrix} \sigma^2_b & \rho_b & \rho_b\\ \rho_b &\sigma^2_b & \rho_b\\ \rho_b & \rho_b &\sigma^2_b \\ \end{pmatrix}A'\Big)}_{\text{component (b)}} $$