If $X$ and $Y$ are two dependent zero-mean normal random variables, does $XY$ has a sub-Gaussian distribution?
My effort: Since $X$ and $Y$ are sub-Gaussians, there exist $c_1,c_2,\alpha_1,\alpha_2>0$ such that $$P(|X|\geq t) \leq c_1 e^{-\alpha_1 t^2} $$ $$P(|Y|\geq t) \leq c_2 e^{-\alpha_2 t^2} $$ We need to prove there exists $c,\alpha>0$ such that $$P(|XY|\geq t) \leq c e^{-\alpha t^2} $$
Since $X, Y$ are not independent and not bivariate, I don't know how to continue from here. I tried to upper bound $P(|XY|\geq t) $ by some function of $P(|X|\geq t)$ and $P(|Y|\geq t)$, but it was not possible. Any idea?