Prohorov inequality

Let {Xi}i=1n be a collectionMathworldPlanetmath of independentPlanetmathPlanetmath random variablesMathworldPlanetmath satisfying the conditions:

a) E[Xi2]< i, so that one can write i=1nE[Xi2]=v2
b) Pr{|Xi|M}=1  i.

Then, for any ε0,

Pr{i=1n(Xi-E[Xi])>ε} exp[-ε2Marsinh(εM2v2)]
Pr{|i=1n(Xi-E[Xi])|>ε} 2exp[-ε2Marsinh(εM2v2)]

(See here ( for the meaning of arsinh(x))

Title Prohorov inequality
Canonical name ProhorovInequality
Date of creation 2013-03-22 16:12:56
Last modified on 2013-03-22 16:12:56
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 17
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 60E15
Synonym Prokhorov inequality