Sub-exponential random variable

A centered random variable is sub-exponential when its moment-generating function is bounded by a Gaussian moment-generating function in a neighbourhood of zero. Equivalently up to universal constants, its tail is bounded by $2e^{-ct/K}$ beyond scale $K$.

A centered random variable $X$ with $\mathbb{E}[X^2]=\nu$ satisfies the Bernstein moment condition with parameter $b$ whenfor every integer $k\ge 2$. Summing its moment-generating series shows that it is sub-exponential with parameters $(2\nu ,2b)$.

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