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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.

Use the convention $\mathbb{E}{e}^{sX}\le e^{{\sigma }^2{s}^2/2}$ for the centered sub-Gaussian random variable. Applying the two-sided Chernoff bound to $X$ gives $\mathbb{P}(|X|>x)\le 2e^{-x^2/(2{\sigma }^2)}$. Since $\mathbb{E}{X}^2\ge 0$, this already supplies the requested bound:For averages of independent random variables, the stronger useful statement is that the centered square of a sub-Gaussian random variable is a sub-exponential random variable. The Bernstein bound for independent sub-exponential variables then give $\mathbb{P}(n^{-1}{\sum }_i(X_i^2-\mathbb{E}{X}_i^2)>t)\le 2\exp [-cn\min \{t^2/{\sigma }^4,t/{\sigma }^2\}]$ for a universal positive $c$. These are the concentration theorems used for quadratic scan statistics.