$X$ and $Y$ are two dependent Gaussian random variables with finite means $\mu_x,\mu_y$, variances $\sigma_x^2,\sigma_y^2$, and covariance $\rho$. Prove $\mathbb{E}[X^2Y^2]$ is finite.
My effort: It is equivalent to show that $\text{Var}[XY]$ is finite. There is a bound on the variance of $XY$ which does not work here since $X$ and $Y$ are not bounded. Another way might be to use this pdf of the product and show that the integral is finite. Although it works for zero means, we can say even if the mean is not zero we are OK. I thought there might be a better solution. Any idea?
Best Answer
I should probably write it as a full answer so that the question is answered: the result immediately follows from Cauchy-Schwartz inequality (as BGM wrote out in the comments) $\mathbb{E}X^2Y^2 \leq \sqrt{\mathbb{E}X^4\mathbb{E}Y^4}$, and for Gaussians the fourth moment is finite.