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 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

[math]\displaystyle{ \sum_{ i = 1 }^n a_i^2 = 1 . }[/math]

Eaton showed that

[math]\displaystyle{ P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \inf_{ 0 \le c \le k } \int_c^\infty \left( \frac{ z - c }{ k - c } \right)^3 \phi( z ) \, dz = 2 B_E( k ) , }[/math]

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

A related bound is Edelman's

[math]\displaystyle{ P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \left( 1 - \Phi\left[ k - \frac{ 1.5 }{ k } \right] \right) = 2 B_{ Ed }( k ) , }[/math]

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

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

[math]\displaystyle{ B_{ EP } = \min\{ 1, k^{ -2 }, 2 B_E \} }[/math]

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

[math]\displaystyle{ \sum_{ i = 1 }^n b_i^2 = 1 . }[/math]

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

[math]\displaystyle{ a_i b_i + \cdots + a_n b_n }[/math]

will converge if and only if

[math]\displaystyle{ \sum_{ i = 1 }^n b_i^2 }[/math]

is finite.

Then

[math]\displaystyle{ E f( a_i b_i + \cdots + a_n b_n ) \le E f( Z ) }[/math]

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

[math]\displaystyle{ E f( 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 } }[/math]

where inf is the infimum.

Let

[math]\displaystyle{ S_n = a_i b_i + \cdots + a_n b_n }[/math]

Then

[math]\displaystyle{ P( S_n \ge x ) \le \frac{ 2e^3 }{ 9 } P( Z \ge x ) }[/math]

The constant in the last inequality is approximately 4.4634.

An alternative bound is also known:

[math]\displaystyle{ P( S_n \ge x ) \le e^{ -x^2 / 2 } }[/math]

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

[math]\displaystyle{ P( | \mu - \sigma | ) \le 0.5 \, }[/math]

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