Eaton's inequality

Let {X_i} be a set of real independent random variables, each with an expected value of zero and bounded above 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

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

Eaton showed that

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

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

A related bound is Edelman's

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

where Φ(x) is cumulative distribution function of the standard normal distribution.

Pinelis has shown that Eaton's bound can be sharpened:[2]

${\displaystyle B_{EP}=\min\{1,k^{-2},2B_{E}\}}$

A set of critical values for Eaton's bound have been determined.[3]

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

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

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

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

will converge if and only if

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

is finite.

Then

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

for f(x) = | x |^p. The case for p ≥ 3 was proved by Whittle[4] and p ≥ 2 was proved by Haagerup.[5]

If f(x) = e^λx with λ ≥ 0 then

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

where inf is the infimum.[6]

Let

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

Then[7]

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

The constant in the last inequality is approximately 4.4634.

An alternative bound is also known:[8]

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

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[9]

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

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