= Centered square of a sub-Gaussian random variable
{title2=$X^2-\mathbb E X^2$}
If $\mathbb E e^{tX}\leq e^{\sigma^2t^2/2}$, 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 $\mathbb P(|X|>x)\leq2e^{-x^2/(2\sigma^2)}$; 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.
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