Eaton's inequality

In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. 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 expected value of zero and bounded by 1 ( | X_i | ≤ 1, for 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

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

where φ(x) is the probability density function of the standard normal distribution.

A related bound is Edelman's

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

where Φ(x) is cumulative distribution function of the standard normal 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 Eaton's bound have been determined.