# Prohorov inequality

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) $\Pr\left\{\left|X_{i}\right|\leq M\right\}=1$  $\forall i$.

Then, for any $\varepsilon\geq 0$,

 $\displaystyle\Pr\left\{\sum_{i=1}^{n}\left(X_{i}-E[X_{i}]\right)>\varepsilon\right\}$ $\displaystyle\leq$ $\displaystyle\exp\left[-\frac{\varepsilon}{2M}\operatorname{arsinh}\left(\frac% {\varepsilon M}{2v^{2}}\right)\right]$ $\displaystyle\Pr\left\{\left|\sum_{i=1}^{n}\left(X_{i}-E[X_{i}]\right)\right|>% \varepsilon\right\}$ $\displaystyle\leq$ $\displaystyle 2\exp\left[-\frac{\varepsilon}{2M}\operatorname{arsinh}\left(% \frac{\varepsilon M}{2v^{2}}\right)\right]$

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

Title Prohorov inequality ProhorovInequality 2013-03-22 16:12:56 2013-03-22 16:12:56 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 17 Andrea Ambrosio (7332) Theorem msc 60E15 Prokhorov inequality