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Eaton's inequality

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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.[1]

Statement of the inequality

Summarize
Perspective

Let {Xi} be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( |Xi | ≤ 1, for 1 ≤ in). The variates do not have to be identically or symmetrically distributed. Let {ai} be a set of n fixed real numbers with

Eaton showed that

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

A related bound is Edelman's[citation needed]

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

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

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

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Summarize
Perspective

Let {ai} be a set of independent Rademacher random variablesP( ai = 1 ) = P( ai = −1 ) = 1/2. Let Z be a normally distributed variate with a mean 0 and variance of 1. Let {bi} be a set of n fixed real numbers such that

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

will converge if and only if

is finite.

Then

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

where inf is the infimum.[6]


Let


Then[7]

The constant in the last inequality is approximately 4.4634.


An alternative bound is also known:[8]

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]

[clarification needed]

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

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References

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