Consider the set of random variables with zero mean and finite second moment. This is a vector space, and $\langle X, Y \rangle = E[XY]$ is a valid inner product on it. Uncorrelated random variables correspond to orthogonal vectors in this space.
(i) Does there exist a similar geometric interpretation for independent random variables in terms of this vector space?
(ii) A collection of jointly Gaussian random variables are uncorrelated if and only if they are independent. Is it possible to give a geometric interpretation for this?