Hoeffding Inequalities (Hoeffding 不等式) 详解

Hoeffding Inequalities (Hoeffding 不等式)

Definition:

Let $\mathbf{X}_1, \cdots , \mathbf{X}_n$ be independent \textbf{bounded} random variables such that $\mathbf{X}_i$ falls in the interval $[a_i, b_i]$ with probability one. Then for any $\mathbf{t>0}$, we have:

$$\begin{equation} \mathbf{P}\lbrace \mathbf{S}_n - \mathbf{E}[\mathbf{S}_n] \geq t \rbrace \leq \exp\left(-\frac{2t^2}{\sum_{i=1}^{n}(b_i-a_i)^2}\right) \end{equation}$$

and

P{ Sn−E[Sn]≤−t}