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Suppose I have transformation defined as $Y_{q\times 1} = C_{q\times p}X_{p\times 1}$, where $X \sim N_p(\mu, \Sigma)$. If $q > p$ how do I compute the distribution of $Y$, since I think the standard result $Y \sim N_q(C\mu, C\Sigma C^T)$ will fail as $C\Sigma C^T$ will be rank deficient. Please help?

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I'm not sure if this answer is satisfying. What you have written I right up to the place where you concluded that the above form cannot represent covariance matrix. Covariance matrix can indeed be not of a full rank.

Look answer of this question for example.

111.

Anyway, you can consider simple example with $p=2$, $q=3$. The last row of $Y$ then can always be represented with first two. (If rows of $C$ are linearly independent.)

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