# Absolute continuity of probability measures determined by dependence structure

We are on $$\mathbb{R}^d$$ with Borel $$\sigma$$-algebra. Let $$\mu_1, ..., \mu_d$$ be probability measures on $$\mathbb{R}$$ and $$\Pi(\mu_1, \mu_2, ..., \mu_d)$$ be the set of probability measures on $$\mathbb{R}^d$$ with first marginal $$\mu_1$$, second marginal $$\mu_2$$, etc.

Given $$\pi^{(1)}, \pi^{(2)} \in \Pi(\mu_1, ..., \mu_d)$$ I am interested in conditions for the absolute continuity $$\pi^{(1)} \ll \pi^{(2)}$$. More specifically, I am looking for sufficient conditions that do not depend on the marginals $$\mu_1, ..., \mu_d$$, but solely on the dependence structures of $$\pi^{(1)}$$ and $$\pi^{(2)}$$.

I am first of all very interested whether these types of conditions or related questions have already been studied.

More specifically, I have the following idea for such a condition, which I could so far neither prove nor disprove:

Let $$C_1, C_2 : [0, 1]^d \rightarrow [0,1]$$ be two copulas. I.e., $$C_1, C_2$$ are CDFs corresponding to $$\nu^{(1)}, \nu^{(2)}$$, which are probability measures on $$[0, 1]^d$$ with uniform marginals.

Let $$\pi^{(1)}, \pi^{(2)} \in \Pi(\mu_1, ..., \mu_d)$$ be given by the dependence structures corresponding to $$C_1, C_2$$, respectively.

Question: If $$\nu^{(1)} \ll \nu^{(2)}$$, does then also hold $$\pi^{(1)} \ll \pi^{(2)}$$?

• If I understand all of your terms correctly, then no...Consider $\nu^1$ which is uniform over all $(x,y)$ where $x$ and $y$ begin with the same digit in base 4, and $\nu^2$ which is similar but with base 2. Then $\nu^1$ is absolutely continuous with respect to $\nu^2$, but this says nothing about the broader dependence structures $\pi$, which may do different things in the null regions of the $\nu$’s. – Matt F. Jan 26 at 2:09
• @MattF. Thanks, this sounds like an interesting example. Do I understand it correctly that $(0.1, 0.5)$ might be a sample of both $\nu^1$ and $\nu^2$ (since 1 mod 4 = 5 mod 4), but $(0.1, 0.3)$ could only a sample of $\nu^2$ but not $\nu^1$? If so, could you describe what kind of structure for $\pi$ you have in mind such that absolute continuity might be lost? – Steve Jan 26 at 10:01