 Product of sub-Gaussian random variables is sub-exponential

If $X$ and $Y$ are 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 $2|XY|\le X^2+Y^2$ converts exponential-square moment bounds for the factors into an exponential moment bound for the product.