Theorem (Prohorov inequality, 1959):

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

a) $E[X_{i}^{2}]<\infty $ $\forall i$ , so that one can write $% \sum_{i=1}^{n}E[X_{i}^{2}]=v^{2}$
b) $\Pr\left\{\left\vert X_{i}\right\vert \leq M\right\} =1$ $\forall i$ .

Then, for any $\varepsilon \geq 0$ , \begin{eqnarray*} \Pr\left\{ \sum_{i=1}^{n}\left( X_{i}-E[X_{i}]\right) >\varepsilon \right\} &\leq &\exp \left[ -\frac{\varepsilon }{2M}\arsinh\left( \frac{\varepsilon M}{2v^{2}}\right) \right] \\ \Pr\left\{ \left\vert \sum_{i=1}^{n}\left( X_{i}-E[X_{i}]\right) \right\vert >\varepsilon \right\} &\leq &2\exp \left[ -\frac{\varepsilon }{2M}\arsinh\left( \frac{\varepsilon M}{2v^{2}}\right) \right] \end{eqnarray*} (See here for the meaning of $\arsinh(x)$ )