Bernstein’s Inequality

Let \(X_1, ..., X_N\) be independent and mean zero, with \(|X_i| < \mu\). Define \(X = \sum_i X_i\). Then, \(\forall \delta > 0\):

\[\mathbb{P}[X \geq \delta] \leq 2 \exp \Big( - \frac{2 \delta^2}{\sum_i \mathbb{E}[X_i^2] - \mu \delta / 3} \Big)\]