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Prove the Prohorov inequality
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Starting from the basic inequality $\exp (-x) \geq 1-x$, it's easy to derive by elementary algebraic manipulations the two inequalities
\begin{aligned} &\exp (x)-x-1 \leq 2(\cosh (x)-1) \\ &2(\cosh (x)-1) \leq x \sinh (x) \end{aligned}
By the Chernoff-Cram&egrave;r bound, we have:
$\operatorname{Pr}\left\{\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)>\varepsilon\right\} \leq \exp \left[-\sup _{t>0}(t \varepsilon-\psi(t))\right]$
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
$E\left[|X|^{k}\right] \leq M^{k} \Longrightarrow E\left[|X|^{k}\right] \leq E\left[X^{2}\right] M^{k-2} \quad \forall k \geq 2, k \in N$
by Platinum (132,156 points)

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