If Co-variance (X, Y) = 0, does this necessarily mean that X and Y are independent?
I have read in previous posts that Co-variance equal to 0 implies that (1) X and Y are independent AND (2) X and Y could be independent, but are not necessarily so. These two interpretations, however, are obviously contradictory.
There was a Soviet paper in the 60's which formally introduced spherically invariant processes. If X(t) is such process, then
X(t) and X(s) are uncorrelated <=====> X(t) and X(s) are independent.
An example of spherically invariant process is multiplicative Gaussian mixture:
X(t) = A * Z(t),
where Z(t) is a Gaussian process and A is some random variable... Generally speaking, 0 correlation does not imply independence. Example:
Y = X * N(0,1),
where X and N(0,1) are independent. Notice that Corr[X,Y] = 0 but X and Y are codependent.
