I've seen a very hand-wavy explanation of how a multivariate normal can be derived from the 1D case, but I wan't really able to find it done rigorously anywhere. Here's an outline of how it went:
- consider two independent random variables $X$ and $Y$ with a gaussian distribution
- because they're independent $P(X, Y) = P(X)P(Y)$
- we can directly write their covariance matrix (since it's diagonal), and we can also write as an outer product
- a correlation between $X$ and $Y$ can be viewed as rotating the joint distribution along the origin
- orthogonal matrix multiplication corresponds to rotation, which means we can (not sure how) find an orthogonal matrix and multiply the covariance matrix with it, which somehow gives the formula for a multivariate normal with an arbitrary covariance matrix
I don't remember how the determinant of the covariance matrix is derived.
Is this just a hand-wavy derivation that is mostly based on intution, or can this be done in a rigorous way to arrive at the actual multivariate normal?