Compatibility Conditions on a Set of Conditional Gaussian Distributions

Suppose that a n-dimensional (random) vector X epsilon R sup n is partitioned into m components as X = (X sub 1,...,X sub m), where the dimension of X sub i is n sub i, i.e., dim(X sub i) = n sub i, for i = 1,...,m and sqrt sup m over sub (i=1) n sub i = n, with m Gaussian conditionals.