Centered square of a sub-Gaussian random variable ($X^2-\mathbb{E}{X}^2$)

If $\mathbb{E}{e}^{tX}\le e^{{\sigma }^2{t}^2/2}$, then $X^2-\mathbb{E}{X}^2$ is a sub-exponential random variable with scale at most a universal constant times ${\sigma }^2$. The Chernoff bound gives $\mathbb{P}(|X|>x)\le 2e^{-x^2/(2{\sigma }^2)}$; integrating this tail bounds $\mathbb{E}|X{|}^{2q}$ by $2(2{\sigma }^2{)}^{q}q!$. Centering preserves a factorial moment bound, so the Bernstein bound for independent sub-exponential variables apply to independent averages of these squares.