Sufficient condtion for the existence of E[|XY|]

Question

I have read a claim that $E[X^4]<\infty$ and $E[Y^4]<\infty$ imply that $E[(XY)^2]<\infty$ but I cannot prove it. $X$ and $Y$ are not independent and they are correlated.

Answer 1

This is an application of Holder's inequality or equivalently Cauchy-Schwarz. Let $\lVert Y \rVert _p=(E|Y|^p)^{1/p}$ for $p\geq 1$. Then $$ E(X^2Y^2)=\lVert X^2Y^2 \rVert _1\leq\lVert X^2\rVert _2\lVert Y^2\rVert _2=(EX^4)^{1/2}(EY^4)^{1/2}<\infty $$ as desired.