Let a selection of random variable $X=(X_1,X_2,\dots, X_n)$ defined on a common background space $(\Omega, \mathcal{F},P)$ be given, and let $F_X:\mathbb{R}^n\to \mathbb{R}$ denote their (joint) CDF.
Let $A\in \text{GL}(n,\mathbb{R})$ be an invertible $n\times n$ matrix over the reals. Which of the following properties can be inferred of $F_{A(X)}$ given that we know them to hold true for $F_X$:
- Absolute continuity (i.e., the distribution $X(P)$ is absolutely continuous wrt. the $n$-dimensional Lebesgue measure $\lambda^n$).
- (Locally) Lipschitz continuity.
- Differentiability.
Im guessing that $1.$ is always true by a simple density transformation. A sufficient condition for the other two to hold true is that $A$ is $\textit{monotone}$, i.e., $Av\geq 0$ implies $v\geq 0$ (with the inequality applying coordinatewise). Indeed, for then $F_{A(X)}=F_X\circ A^{-1}$ with properties $1.-3.$ jumping out immediately. Is this condition of monotonicity also necessary?
Can the overarching question on permanence properties be addressed by other means?