Let $f_{X,Y}(x,y)$ be the joint probability density of correlated random variables $X$ and $Y$ based on a Copula $C$ (Gaussian in my case) where $f_X(x)$ and $f_Y(y)$ are the marginal probability density functions of $X$ and $Y$ respectively. Similarly, $F_X(x)$ and $F_Y(y)$ are the marginal cumulative density functions of $X$ and $Y$ respectively. The marginal pdfs are not Gaussian and with different support ($X\in\mathbb{R}$ and $Y\in [a,b]$)
The joint probability density can be expressed as \begin{equation} f_{X,Y}(x,y)= c(w,z)\,f_X(x)\,f_Y(y), \end{equation} where $c(w,z)$ is the bivariate Gaussian copula density.
Now I define two new variables $U$ and $V$ obtained by a linear combination of $X$ and $Y$, that is \begin{equation} \left(\begin{array}{x} U\\V \end{array}\right) =A \left(\begin{array}{x} X\\Y \end{array}\right), \end{equation} where $A$ is a $2\times 2$ matrix.
How can I obtain the joint pdf of the new variables $f_{U,V}(u,v)$?
My attempt is to use the Jacobian transformation: \begin{equation} f_{U,V}(u,v)=\frac{1}{|J|}f_{X,Y}(x(u,v),y(u,v)) \end{equation} where $|J|$ is the determinant of the jacobian $|J|=A_{11}A_{22}-A_{12}A_{21}$
While this works for a bivariate Gaussian joint PDF, it does not work for the copula.
Any help in this matter would be very welcome.
