Question

. Let X and Y be square-integrable random variables, and define the L 2

-norm of X to be ∥X∥ 2=def E{X 2} 1/2
. Prove Minkowski's inequality: ∥X+Y∥ 2​≤∥X∥ 2+∥Y∥ 2
​
. [Hint: Compute E{(X+Y)
2
} and use the Schwarz inequality on the cross term. Then rewrite it in terms of norms.]

Answer 1

To prove Minkowski's inequality for L2 norms, we compute E{(X+Y)^2} and apply the Schwarz inequality to the cross term. This results in ∥X+Y∥2 ≤ ∥X∥2 + ∥Y∥2.

Step-by-step explanation:

To prove Minkowski's inequality ∥X+Y∥2 ≤ ∥X∥2 + ∥Y∥2,

we begin by computing E{(X+Y)2}. Using the linearity of expectation, we have:

E{(X+Y)2} = E{X2} + 2E{XY} + E{Y2}

Next, we apply the Schwarz inequality to the cross term E{XY}. The Schwarz inequality states that for any two square-integrable random variables A and B, |E{AB}| ≤ ∥A∥2 * ∥B∥2. Therefore, |E{XY}| ≤ ∥X∥2 * ∥Y∥2.

Substituting this inequality into our expression for E{(X+Y)2}, we get:

E{(X+Y)2} ≤ E{X2} + 2∥X∥2 * ∥Y∥2 + E{Y2}

We can rewrite this expression in terms of norms as:

∥X+Y∥2 ≤ ∥X∥2 + ∥Y∥2