I have a few questions with respect to the gaussian distribution, its moments and independence.
So a gaussian distribution is fully specified by its first two moments, the mean and variance (or covariance matrix in the multivariate case). Since uncorrelatedness means unrelated in the first two statistical moments and independence means unrelatedness in all higher moments as well, the jointly gaussian distributed variables are independent if they re uncorrelated.
My question now is, what does it mean that it is fully specified by its first two moments? I mean the higher order moments are not zero? Can someone maybe also give a counter example, so how it looks if a distribution is not fully specified by its first two moments?
And another questions relates to the rotational invariance of the gaussian. For independent component analysis one demands that the latent variables are not gaussian distributed because the gaussian can -apparently - only be defined up to a rotation. What exactly does that mean and does it have anything to do with its property that it is fully specified by its first two moments?