= Bernstein bound for independent sub-exponential variables
{c}
{title2=$\mathbb P(|\sum_iX_i|\geq t)\leq2e^{-t^2/(2(v+bt))}$}
Suppose independent centered <sub-exponential random variables> satisfy $\log\mathbb E e^{\lambda X_i}\leq v_i\lambda^2/(2(1-b|\lambda|))$ for $|\lambda|<1/b$, and put $v=\sum_iv_i$. 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 $v_i\leq CK^2$ and $b\leq CK$, giving $\mathbb P(|n^{-1}\sum_iX_i|>t)\leq2e^{-c n\min\{t^2/K^2,t/K\}}$. This form of <Bernstein's inequality> allows unbounded variables with factorial moment control, including the <centered square of a sub-Gaussian random variable>.
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