Bernstein's Inequality
Let $X_1, \ldots, X_n$ be independent with common mean $0$, variances $\mathbb{V}(X_i) \leq \sigma^2$, and suppose that $|X_i| \leq M$ for all $i$.
Then, for every $\epsilon > 0$,
\begin{align} \mathbb{P}(|\overline{X}| \geq \epsilon) \leq 2 e^{-c \, n \epsilon^2} \end{align}
where
(2)\begin{align} \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i \end{align}
and
(3)\begin{align} \frac{1}{c} = 2 \sigma^2 + \frac{2}{3}M \epsilon. \end{align}
page revision: 0, last edited: 05 Nov 2015 01:06