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by Platinum
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State the Prohorov inequality

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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 \(\varepsilon \geq 0\),
\[
\begin{gathered}
\operatorname{Pr}\left\{\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)>\varepsilon\right\} \leq \exp \left[-\frac{\varepsilon}{2 M} \operatorname{arsinh}\left(\frac{\varepsilon M}{2 v^{2}}\right)\right] \\
\operatorname{Pr}\left\{\left|\sum_{i=1}^{n}\left(X_{i}-E\left[X_{i}\right]\right)\right|>\varepsilon\right\} \leq 2 \exp \left[-\frac{\varepsilon}{2 M} \operatorname{arsinh}\left(\frac{\varepsilon M}{2 v^{2}}\right)\right]
\end{gathered}
\]
by Platinum
(106,844 points)

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