Prohorov inequality

Theorem (Prohorov inequality, 1959):

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

a) $\forall i$, so that one can write ${\sum }_{i=1}^{n}E[X_i^2]=v^2$
b) $\mathrm{Pr}\left\{\left|X_i\right|\le M\right\}=1$  $\forall i$.

Then, for any $\epsilon \ge 0$,

$\mathrm{Pr}\left\{{\displaystyle \sum _{i=1}^n}\left(X_i-E[X_i]\right)>\epsilon \right\}$$\le$$\exp \left[-{\displaystyle \frac{\epsilon }{2M}}\mathrm{arsinh}\left({\displaystyle \frac{\epsilon M}{2v^2}}\right)\right]$

$\mathrm{Pr}\left\{\left|{\displaystyle \sum _{i=1}^n}\left(X_i-E[X_i]\right)\right|>\epsilon \right\}$$\le$$2\exp \left[-{\displaystyle \frac{\epsilon }{2M}}\mathrm{arsinh}\left({\displaystyle \frac{\epsilon M}{2v^2}}\right)\right]$

(See here (http://planetmath.org/AreaFunctions) for the meaning of $\mathrm{arsinh}(x)$)