# arrow_back State the Prohorov inequality

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State the Prohorov inequality

Theorem (Prohorov inequality, 1959$)$ :
Let $\left\{X_{i}\right\}_{i=1}^{n}$ be a collection of independent random variables satisfying the conditions:
a) $E\left[X_{i}^{2}\right]<\infty \forall i$, so that one can write $\sum_{i=1}^{n} E\left[X_{i}^{2}\right]=v^{2}$
b) $\operatorname{Pr}\left\{\left|X_{i}\right| \leq M\right\}=1 \quad \forall i$
Then, for any $\varepsilon \geq 0$,
$\begin{gathered} \operatorname{Pr}\left\{\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)>\varepsilon\right\} \leq \exp \left[-\frac{\varepsilon}{2 M} \operatorname{arsinh}\left(\frac{\varepsilon M}{2 v^{2}}\right)\right] \\ \operatorname{Pr}\left\{\left|\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)\right|>\varepsilon\right\} \leq 2 \exp \left[-\frac{\varepsilon}{2 M} \operatorname{arsinh}\left(\frac{\varepsilon M}{2 v^{2}}\right)\right] \end{gathered}$
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