Probability over a subset

Any subset of variables obtained from linear combinations of Gaussian variables is Gaussian for the following reason. Let us consider independent linear combinations z. of the x (with a = 1.M and M N) 

Let us try to determine the probability law Q(zt. zu) associated with these variables. The corresponding characteristic function is 

The characteristic function is also the generating function of mean values. By expanding eqn (A. 11), we find 

This quantity is real because the mean values corresponding to an uneven number of variables vanish. In particular, by expanding (A.14) with respect to the kj and by identifying the expansion with (A.20), we find 

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