Bennett inequality

Theorem:(Bennett inequality, 1962):

Let ${\{X_i\}}_{i=1}^n$ be a collection of independent randomvariables satisfying the conditions:

a) $\forall i$, so that one can write ${\sum }_{i=1}^{n}E[X_i^2]=v^2$
b) $\mathrm{Pr}\left\{\left|X_i\right|\le M\right\}=1$  $\forall i$.

Then, for any $\epsilon \ge 0$,

$$\mathrm{Pr}\left\{\sum _{i=1}^n\left(X_i-E[X_i]\right)>\epsilon \right\}\le \exp \left[-\frac{v^2}{M^2}\theta \left(\frac{\epsilon M}{v^2}\right)\right]\le \exp \left[-\frac{\epsilon }{2M}\ln \left(1+\frac{\epsilon M}{v^2}\right)\right]$$

where

$$\theta \left(x\right)=\left(1+x\right)\ln \left(1+x\right)-x$$

Remark:Observing that $\left(1+x\right)\ln \left(1+x\right)-x\ge 9\left(1+\frac{x}{3}-\sqrt{1+\frac{2}{3}x}\right)\ge \frac{3x^2}{2\left(x+3\right)}$ $\forall x\ge 0$, and plugging these expressions into thebound, one obtains immediately the Bernstein inequality under the hypotheses ofboundness of random variables, as one might expect. However, Bernsteininequalities, although weaker, hold under far more general hypotheses thanBennett one.