Correlation and squared variables

Question

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.

Answer 1

If $X$ and $Y$ are independent random variables, then so are $g(X)$ and $h(Y)$ independent random variables for all (measurable) functions $g(\cdot)$ and $h(\cdot)$. Thus, if $X$ and $Y$ are independent (and hence uncorrelated), then $X^2$ and $Y^2$ cannot be correlated random variables; they too must be independent (and hence uncorelated) random variables. Now, since $$X ~\text{and}~ Y ~ \text{independent} \Rightarrow X^2 ~\text{and}~ Y^2 ~ \text{independent}$$ it follows that $$X^2 ~\text{and}~ Y^2 ~ \text{not independent} \Rightarrow X ~\text{and}~ Y ~ \text{not independent}.$$ Since $X^2$ and $Y^2$ are correlated, they are not independent, and so $X$ and $Y$ are not independent either. Whether $X$ and $Y$ are correlated or not has no bearing on the matter.

Answer 2

Normal distribution doesnt explicitely imply independence. If two jointly normally distributed random variables are uncorrelated, then they are independent.

In your case you know that two random variables $X$ and $Y$ are dependent and uncorrelated. This means

$$P(X=a|X=b)\neq P(X=a)$$

for at least one $a$ and $b$ and $$E[XY]=E[X]E[Y]$$

Given these two conditions, extra conditions defined on $X^2$ or $Y^2$ will not change the independency condition.