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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 $\pi^{(1)} \ll \pi^{(2)}$?

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  • $\begingroup$ 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. $\endgroup$ – Matt F. Jan 26 at 2:09
  • $\begingroup$ @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? $\endgroup$ – Steve Jan 26 at 10:01

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