TITLE: A geometric interpretation of independence?
QUESTION [18 upvotes]: 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. 
Questions:
(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?

REPLY [5 votes]: If you leave the realm of abstract probability spaces and focus on probability in Banach spaces, there's a lot of geometry to take advantage of.  Here's an example.
Let $X$ be a Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ such that continuous linear functionals are square-integrable (i.e. $\int_X |f(x)|^2 ~d\mathbb P(x) < \infty$ for all $f \in X^*$).  For example, $X = C([0,1])$ with Wiener measure $\mathbb P$.
These are sufficient conditions for there to exist a mean $m \in X$ and covariance operator $K : X^* \to X$ such that $$\mathbb Ef = f(m) \qquad \mathrm{and} \qquad \mathbb E (fg) - f(m)g(m) = f(Kg)$$
for all $f, g \in X^*$.  One can show that
$$\mathbb P \left( m + \overline{KX^*} \right) = 1.$$
Under these very general assumptions, the probability concentrates on the affine subspace generated by the mean and covariance.
Reference:  Vakhania, Tarieladze and Chobanyan, Probability Distributions in Banach Spaces