I was looking into ICA, I have read the whole article Independent component analysis: Algorithms and applications. But I think I could not find the answer that why non-Gaussian Variables are independent.

Any Enlightenment Please!

I have read a lot about ICA. I think I could not find the answer that why non-Gaussian Variables are independent. What I understand, Central Limit Theorem states that

distribution of sum of independent variable tends toward more Gaussian than its original random variables.

$ s_i $ are the number of original independent sources in ICA; whereas ICA model is $x=As$. So we can define $y=w^Ts$. The main goal is to find the unmixing matrix $w$ that maximize the non-Gausaanity. So my question is what is non-Gaussanity here and why its necessary to maximize it to extract the original sources.

Any Enlightenment Please!