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 Morris L. 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.

Related inequalities

Let a_i be a set of independent Rademacher random variables – P( a_i = 1 ) = P( a_i = −1 ) = 1/2. Let Z be a normally distributed variate with a mean 0 and variance of 1. Let b_i be a set of n fixed real numbers such that

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

This last condition is required by the Riesz–Fischer theorem which states that that

$a_{i}b_{i}+\cdots +a_{n}b_{n}$

will converge if and only if

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

is finite.

Then

$Ef(a_{i}b_{i}+\cdots +a_{n}b_{n})\leq Ef(Z)$

for f(x) = | x |^p. The case for p ≥ 3 was proved by Whittle and p ≥ 2 was proved by Haagerup.

If f(x) = e^λx with λ ≥ 0 then

$Ef(a_{i}b_{i}+\cdots +a_{n}b_{n})\leq \inf \left[{\frac {E(e^{\lambda Z})}{e^{\lambda x}}}\right]=e^{-x^{2}/2}$

where inf is the infimum.

Let

$S_{n}=a_{i}b_{i}+\cdots +a_{n}b_{n}$

Then

$P(S_{n}\geq x)\leq {\frac {2e^{3}}{9}}P(Z\geq x)$

The constant in the last inequality is approximately 4.4634.

An alternative bound is also known:

$P(S_{n}\geq x)\leq e^{-x^{2}/2}$

This last bound is related to the Hoeffding's inequality.

In the uniform case where all the b_i = n^−1/2 the maximum value of S_n is n^1/2. In this case van Zuijlen has shown that

$P(|\mu -\sigma |)\leq 0.5\,$

where μ is the mean and σ is the standard deviation of the sum.