Check whether Normal Mixture Margins are Independent

Question

Consider a $W \sim \text{Pareto}(\beta)$ and let $\mathbf{X}=(X_1,X_2)^T$ be a bivariate vector whose components are given by:

$$X_1 = \sqrt{W}(Z_1+Z_2) \quad \text{and} \quad X_2 = \sqrt{W}(Z_1-Z_2)$$

where $Z_1$ and $Z_2$ iid standard normal, independent of $W$.

Now since it is obvious that the margins of $\mathbf{X}$ are not independent, I would like to show that formally. Natural way of showing this would be to show that $\mathrm{E}\left[X_1X_2\right] \neq \mathrm{E}[X_1]\mathrm{E}[X_2]$, however this approach doesn't produce anything useful.

How does one go about proving random variables are $\textit{not}$ independent in general?

Showing that their joint PDF/CDF do not factorize into a product form seems difficult in general.

Answer 1

Partial answer: compute $EX_1^{2}X_2^{2}$ and $EX_1^{2}EX_2^{2}$. If these two are equal you will see that $EW^{2}=(EW)^{2}$ or $var (W)=0$ which is not true. However this works only for $\beta > 2$ since $W$ has infinite variance for $\beta \leq 2$.

Answer 2

Hint:

If $X,Y$ are independent then $X^2,Y^2$ are independent.

or more generally then $f(X),f(Y)$ are independent (for suitable $f$).