# arrow_back Prove the Bennett inequality

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Prove the Bennett inequality

By Chernoff-Cram&egrave;r inequality, we have:
$\operatorname{Pr}\left\{\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)>\varepsilon\right\} \leq \exp \left[-\sup _{t \geq 0}(t \varepsilon-\psi(t))\right]$
where
$\psi(t)=\sum_{i=1}^{n}\left(\ln E\left[e^{t X_{i}}\right]-t E\left[X_{i}\right]\right)$
Keeping in mind that the condition
$\operatorname{Pr}\left\{\left|X_{i}\right| \leq M\right\}=1 \quad \forall i$
implies that, for all $i$,
$E\left[\left|X_{i}\right|^{k}\right] \leq M^{k} \quad \forall k \geq 0$
and since $\ln x \leq x-1 \forall x>0$, and
$E\left[|X|^{k}\right] \leq M^{k} \quad \Longrightarrow \quad E\left[|X|^{k}\right] \leq E\left[X^{2}\right] M^{k-2} \quad \forall k \geq 2, k \in N$
by Platinum
(106,844 points)

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