Distribution function for $f: \mathbb{R}^m \to \mathbb{R}^n$ where $m < n$

Question

Suppose $X$ is a random variable in $\mathbb{R}^n$ and $Y = f(X)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$. $Y$ is a random variable as well and the probability distributions for $X$ and $Y$ are related by

$$ p_X(x) = \frac{p_{Y}(y)}{\left| det \frac{\partial(y_1,\ldots,y_n)}{\partial(x_1,\ldots,x_n)} \right|} $$

Is there a similar result when $m \neq n$? more specifically $m < n$ (strictly less)?

Answer 1

If $m < n$ and $f$ is everywhere differentiable, then measure of $f(\mathbb R ^ m)$ is zero (see, for example, this answer - we can extend $f$ to $\mathbb R^n$ to apply it directly by using $g(x_1, \ldots, x_n) = f(x_1, \ldots, x_m)$ - then $\det D g$ is zero everywhere). So support of $f(X)$ has zero measure and thus $f(X)$ doesn't have density.

Answer 2

I don't know the answer, so this is just a long comment.

If I understand correctly, your formula only works if $f$ is bijective. I also think we need for $f$ be (continuously?) differentiable with non-zero derivative everywhere. If someone could comment on these requirements, that would be great.

We could try firstly weakening this to allow $f$ to be a surjective continuously differentiable function. In this case, we expect that $p_Y(y)$ should be a sum over preimages of $y$. Like so:

$$p_{f(X)}(y) = \sum_{x \in f^{-1}(y)} p_X(x) \cdot |\det f'(x)|$$

For a function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$, I expect that the appropriate variant on this would usually be some kind of an integral. Something like $$p_{f(X)}(y) = \int_{x \in f^{-1}(y)} p_X(x) \cdot |\det f'(x)|$$

with possibly some further subtleties involved.

Another direction this might be generalized is to allow $f$ to be injective but not necessarily surjective. This is to cover cases like $f : \mathbb{R} \rightarrow \mathbb{R}^2$. The notion of a Hausdorff measure seems relevant. For example, it might be possible to get a density function for $f(X)$ not with respect to $H^2_{\mathbb{R}^2}$ (the Lebesgue measure) but with respect to $H^1_{\mathbb{R}^2}$. In the comments, drhab suggests a different and more technical proposal of using "local Lebesgue measures". I'm not qualified to comment on such things, unfortunately, but that might be worth reading about.