Uniqueness of real and complex linear independent component analysis revisited

Comon showed using the Darmois-Skitovitch theorem that under mild assumptions a real-valued random vector and its linear image are both independent if and only if the linear mapping is the product of a permutation and a scaling matrix. In this work, a much simpler, direct proof is given for this theorem and generalized to the case of random vectors with complex values. The idea is based on the fact that a random vector is independent if and only if locally the Hessian of its logarithmic density is diagonal.