I came across the following inequality in one of my calculations ($ X,Y$ are centered random variables): $ $ \operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(XY)^2$ $ or, written in terms of covariances, $ $ \operatorname{Cov}(X^2,Y^2) \geq 2 \operatorname{Cov}(X,Y)$ $ . If $ (X,Y)=(U,V)$ is a two-dimensional centered Gaussian, this becomes an equality and if $ (X,Y)=(H_p(U),H_q(V))$ ,Read more