According to my textbooks if two variables are uncorrelated, they are not necessarily independent (unless they are normally distributed).
My question is, are 2 variables still not independent if they are not correlated, but their squares are correlated? I believe that they are still not independent, but I am not sure since at some point I thought that what if the squares of those 2 variables have Chi-distribution (form a new varialbe Y) and then the variables are normally distributed and independent. So I am quite confused now.
I would be very grateful to you for your help. Thank you very much.