Bennett 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,




Remark: Observing that (1+x)ln(1+x)-x9(1+x3-1+23x)3x22(x+3) x0, and plugging these expressions into the bound, one obtains immediately the Bernstein inequality under the hypotheses of boundness of random variables, as one might expect. However, Bernstein inequalities, although weaker, hold under far more general hypotheses than Bennett one.

Title Bennett inequality
Canonical name BennettInequality
Date of creation 2013-03-22 16:12:25
Last modified on 2013-03-22 16:12:25
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 10
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 60E15