Product of sub-Gaussian random variables is sub-exponential
= Product of sub-Gaussian random variables is sub-exponential
If $X$ and $Y$ are <sub-Gaussian random variable>[sub-Gaussian], without requiring <independence> between them, then $XY-\mathbb E(XY)$ is a <sub-exponential random variable>; equivalently, $XY$ has a finite exponential absolute <moment> at some positive scale. The inequality $2|XY|\leq X^2+Y^2$ converts exponential-square <moment> bounds for the factors into an exponential <moment> bound for the product.