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.[1]
Let be a set of real independent random variables, each with an expected value of zero and bounded above by 1 (|Xi | ≤ 1, for 1 ≤ i ≤ n). The variates do not have to be identically or symmetrically distributed. Let be a set of n fixed real numbers with
n | |
\sum | |
i=1 |
2 | |
a | |
i |
=1.
Eaton showed that
P\left(\left|
n | |
\sum | |
i=1 |
aiXi\right|\gek\right)\le2inf
infty | |
\int | |
c |
\left(
z-c | |
k-c |
\right)3\phi(z)dz=2BE(k),
where φ(x) is the probability density function of the standard normal distribution.
A related bound is Edelman's
P\left(\left|
n | |
\sum | |
i=1 |
aiXi\right|\gek\right)\le2\left(1-\Phi\left[k-
1.5 | |
k |
\right]\right)=2B(k),
where Φ(x) is cumulative distribution function of the standard normal distribution.
Pinelis has shown that Eaton's bound can be sharpened:[2]
B=min\{1,k,2BE\}
A set of critical values for Eaton's bound have been determined.[3]
Let be a set of independent Rademacher random variables – P(ai = 1) = P(ai = −1) = 1/2. Let Z be a normally distributed variate with a mean 0 and variance of 1. Let be a set of n fixed real numbers such that
n | |
\sum | |
i=1 |
2 | |
b | |
i |
=1.
This last condition is required by the Riesz–Fischer theorem which states that
aibi+ … +anbn
will converge if and only if
n | |
\sum | |
i=1 |
2 | |
b | |
i |
is finite.
Then
Ef(aibi+ … +anbn)\leEf(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) = eλx with λ ≥ 0 then
Ef(aibi+ … +anbn)\leinf\left[
E(e) | |
e |
\right]=
-x2/2 | |
e |
Let
Sn=aibi+ … +anbn
Then[7]
P(Sn\gex)\le
2e3 | |
9 |
P(Z\gex)
The constant in the last inequality is approximately 4.4634.
An alternative bound is also known:[8]
P(Sn\gex)\le
-x2/2 | |
e |
This last bound is related to the Hoeffding's inequality.
In the uniform case where all the bi = n−1/2 the maximum value of Sn is n1/2. In this case van Zuijlen has shown that[9]
P(|\mu-\sigma|)\le0.5
where μ is the mean and σ is the standard deviation of the sum.