Eaton's inequality

Eaton's inequality is a bound on the maximal values of a linear combination of bounded random variables.

History

This inequality was described in 1974 by Eaton.

Statement of the inequality

Let X_i be a set of real independent random variables each with a mean of zero and bounded by 1 ( | X_i | ≤ 1). Let 1 ≤ i ≤ n. The variates do not have to be identically or symmetrically distributed. Let a_i be a set of n fixed real numbers with

$\sum _{i=1}^{n}a_{i}^{2}=1$

Eaton showed that

$P(|\sum _{i=1}^{n}a_{i}X_{i}|\geq k)\leq 2\inf _{0\leq c\leq k}\int _{c}^{\infty }({\frac {z-c}{k-c}})^{3}\phi (z)dz=2B_{E}(k)$

where φ( x ) is the normal probability density.

A related bound is Edelman's

$P(|\sum _{i=1}^{n}a_{i}X_{i}|\geq k)\leq 2(1-\Phi [k-{\frac {1.5}{k}}])=2B_{Ed}(k)$

where Φ( x ) is the normal probability distribution.

Pinelis has shown that Eaton's bound can be sharpened:

$B_{EP}=min(1,k^{-2},2B_{E})$

A set of critical values for this bound have been determined by Dufour and Hallin.