Hello http://www.mymathforum.com/images/sm...icon_smile.gif.

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.

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- Sep 11th 2010, 08:00 AMartnoageProbability-measure theory question
Hello http://www.mymathforum.com/images/sm...icon_smile.gif.

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. - Sep 11th 2010, 01:14 PMMoo
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 :D

I don't think it's easy to characterize, there is already such a wide range of possible marginal measures... - Sep 11th 2010, 01:27 PMartnoage
You cant imagine how much helpful that was. It would be great if i could have some informations in the more general case but still this is enough for what i am doing at the moment. Kolmogorov Bless u :).

- Sep 11th 2010, 01:32 PMMoo
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 ? - Sep 11th 2010, 01:44 PMartnoage
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 :(.

- Sep 11th 2010, 01:54 PMMoo
Yes, I'm afraid I was completely wrong in my first post... which I wrote partly joking because I was pretty sure it couldn't help you at all... I'm sincerely sorry :(

- Sep 11th 2010, 01:59 PMartnoage
Np. You still gave me some ideas though. Thanks for replying anyway :)