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. 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?