Product of sub-Gaussian random variables is sub-exponential

If $X$ and $Y$ are sub-Gaussian, without requiring independence between them, then $XY$ is sub-exponential. 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.

For independent identically distributed pairs $(U_i,V_i)$ whose coordinates are sub-Gaussian with parameter $\sigma /4$, Bernstein's inequality applied to the centered products gives