 Sub-Gaussian deviation bound

A sub-Gaussian deviation bound controls tails through a Gaussian-shaped exponential. If a centered random variable $X$ satisfies $\mathbb{E}{e}^{tX}\le e^{{\sigma }^2{t}^2/2}$ for every real $t$, with $\sigma >0$, then for $u>0$,Indeed, exponential Markov's inequality gives $\mathbb{P}(X\ge u)\le \exp (-tu+{\sigma }^2{t}^2/2)$; 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 $\mathbb{P}(|\overline{X}|\ge u)\le 2e^{-n{u}^2/(2{\sigma }^2)}$. The proxy need not equal the actual variance, and dependence requires a separate argument rather than automatic multiplication of moment-generating functions.