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Suppose independent centered sub-exponential random variables satisfy $\log \mathbb{E}{e}^{\lambda X_i}\le v_i{\lambda }^2/(2(1-b|\lambda |))$ for $|\lambda |<1/b$, and put $v={\sum }_i{v}_i$. Independence adds the log moment-generating functions. The Chernoff bound with $\lambda =t/(v+bt)$ gives the displayed two-sided tail. If each variable has common scale $K$, one may take $v_i\le C{K}^2$ and $b\le CK$, giving $\mathbb{P}(|n^{-1}{\sum }_i{X}_i|>t)\le 2e^{-cn\min \{t^2/K^2,t/K\}}$. This form of Bernstein's inequality allows unbounded variables with factorial moment control, including the centered square of a sub-Gaussian random variable.

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.