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Given a random vector X with the multivariate normal distribution F(X), we know that, for two vectors a and b, the projections $A=\sum_j a_j X_j $ and $B=\sum_i b_i X_i $ are univariate normal.

I'm interested in the joint distribution of A and B. Is their joint distribution normal? Is the dependence between A and B described only by their correlation? (do they have only linear dependence?) Thank you for any insight. References are highly appreciated as well.

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I found the answer for the above question and I thought is nice to share it. So the answer is yes: A and B are joint normal and so the relation between them is determined by the correlation. This is due to the properties of the characteristic function of a multivariate normal. – KAT Aug 4 '13 at 11:54

The vector (A,B) consisting of two scalars is a "two-dimensional projection" of the multivairate normal vector X. More generally, if you multiply X with any matrix such that you get a new vector with dimension <= than the original dimension of X, you again get a multivariate normal vector. That is, all the components of the new vector will be normal, and their dependence is determined only by their correlations.

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Yes, @Leonid indeed. It would of been nice if you could give the answer before I did. See above. – KAT Aug 4 '13 at 17:51
The condition that the "new vector (has) dimension <= than the original dimension" is unnecessary and irrelevant. – Did Nov 7 '13 at 10:51

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