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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

$P(\left|\sum _{i=1}^n{a}_i{X}_i\right|\ge k)\le 2{\inf }_{0\le c\le k}\underset{c}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{z-c}{k-c}\right)}^3\varphi (z)dz=2B_E(k),$

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

A related bound is Edelman'sTemplate:Cn

$P(\left|\sum _{i=1}^n{a}_i{X}_i\right|\ge k)\le 2(1-\mathrm{\Phi }[k-\frac{1.5}{k}])=2B_{Ed}(k),$

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

$Ef(a_i{b}_i+\cdots +a_n{b}_n)\le Ef(Z)$

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

$Ef(a_i{b}_i+\cdots +a_n{b}_n)\le \inf \left[\frac{E(e^{\lambda Z})}{e^{\lambda x}}\right]=e^{-x^2/2}$

where inf is the infimum.

Let

$S_n=a_i{b}_i+\cdots +a_n{b}_n$

Then

$P(S_n\ge x)\le \frac{2e^3}{9}P(Z\ge x)$

The constant in the last inequality is approximately 4.4634.

An alternative bound is also known:

$P(S_n\ge x)\le e^{-x^2/2}$

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

$P(|\mu -\sigma |)\le 0.5$Template:Clarification needed

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

References

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