I've been reading about Independent Component Analysis and the FastICA algorithm. The wiki page for FastICA states:
FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology.[1][2] Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify.
I understand that non-gaussianity can serve as a proxy for independence due to the central limit theorem, but why is it that independence itself requires infinite data to verify?