Hello .
Does anyone know any way that you can characterize the set (or any nontrivial subset) of the probability measures in R^{2} with fixed marginals?
Thanks for your time.
If the marginals have the normal (gaussian) density wrt Lebesgue's measure, it's easy since we know the pdf of a bivariate gaussian distribution
I don't think it's easy to characterize, there is already such a wide range of possible marginal measures...
Huh ?
How did I help you exactly ? I just said that since we know the pdf (hence the probability measure) of (X,Y), a Gaussian vector (not just the joint distribution), we know the marginals of X and Y and can characterize the set of probability measures such that this stuff is verified.
But it is just a tiny point in an ocean of measures...
What exactly were you looking for ? I mean why did you ask such a question ?
It is quite complicated to explain what i am doing. But my first toy model has Gaussians as marginals, and i can test there if my assumption is true or not. (if my assumption doesnt even work with gaussians then it has no meaning working with something else). So i thought that what you said, is that when my marginals are Gaussian then always the joint measure is a bivariate Gaussian dist. But probably you meant the other way around .