= Sub-Gaussian deviation bound
A <sub-Gaussian deviation bound> controls tails through a Gaussian-shaped exponential. If a centered <random variable> $X$ satisfies $\mathbb Ee^{tX}\leq e^{\sigma^2t^2/2}$ for every real $t$, with $\sigma>0$, then for $u>0$,
$$
\mathbb P(|X|\geq u)\leq2e^{-u^2/(2\sigma^2)}.
$$
Indeed, exponential <Markov's inequality> gives $\mathbb P(X\geq u)\leq\exp(-tu+\sigma^2t^2/2)$; minimizing at $t=u/\sigma^2$ and applying the argument to $-X$ proves the result.
<Independent> centered observations with a common <variance> proxy $\sigma^2$ therefore satisfy $\mathbb P(|\overline X|\geq u)\leq2e^{-nu^2/(2\sigma^2)}$. The proxy need not equal the actual <variance>, and dependence requires a separate argument rather than automatic multiplication of <moment-generating functions>.
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