Let X,Y and $\epsilon$ be three independent random variables with X,Y satisfying the standard normal distribution $\mathcal{N}(0,1)$ and $\mathbb{P}(\epsilon=-1)=\mathbb{P}(\epsilon=1)=\frac12$, then is $(\epsilon|X|,\epsilon|Y|)$ a Gaussian vector?
I can justify that $\epsilon|X|,\epsilon|Y|$ also satisfy the standard normal distribution $\mathcal{N}(0,1)$, however, they are not independent. And their covariance does not equal to zero too. I really wonder how to judge this vector, so I would appreciate it if someone can explain this to me. Thank you so much!
Of course the random vector $(\epsilon|X|,\epsilon|Y|)$ is not a Gaussian vector since it has density equal to $$\frac{1}{\pi}e^{-\frac{1}{2}(x^2+y^2)}1_{xy>0}$$ but it is not concentrated on the whole $R^2.$