Note on the Grüss Inequality

Let (Ω, F, P) be a probability space, and let X : Ω → R be a bounded random variable with

(1)  m_{X} := ess inf X,  M_{X} := ess sup X.

Proposition.

(2)  |Cov(X, Y)| ≤ \frac{(M_{X} − m_{X})(M_{Y} − m_{Y})}{4}.

Proof. The general case follows immediately from the special case X = Y and the Cauchy-Schwarz inequality. Thus we shall assume that X = Y and show that

(3)  Var(X) ≤ \frac{(M_{X} − m_{X})^{2}}{4} .

In proving (3) we first assume that X has the following additional symmetry properties:

(4)  E(X) = 0 and m_{X} = −M_{X}.

Then

(5)  Var(X) = E(X^{2}) ≤ M_{X}^{2} = [(M_{X} − m_{X})/2]^{2} = \frac{(M_{X} − m_{X})^{2}}{4} .

In general, apply the preceding to X^{∗} := Z(X − c), where c := (M_{X} + m_{X})/2 and Z is a random variable independent of X with P[Z = 1] = P[Z = −1] = 1/2. Clearly M_{X^{∗}} = −m_{X^{∗}} = (M_{X} − m_{X})/2, and

(6)  Var(X) ≤ E [(X − c)^{2}] = Var(X^{∗}) ≤ \frac{(M_{X^{∗}} − m_{X^{∗}})^{2}}{4} = \frac{(M_{X} − m_{X})^{2}}{4} .

It is evident from (5) and (6) that equality holds in (3) if and only if

(7)  P [X^{∗} = M_{X^{∗}}] = P [X^{∗} = m_{X^{∗}}] = \frac{1}{2},

or what amounts to the same thing

(8)  P [X ∈ {M_{X}, m_{X}}] = 1.

That is, equality holds in (3) if and only if the essential range of X contains at most two points.

References

[1] X. Li, R.N. Mohapatra, R.S. Rodriguez: Grüss-type inequalities, J. Math. Anal. Appl., 267 (2002) 434–443.