Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

F.2 TRIANGLE AND MINKOWSKI INEQUALITIES

Theorem F.3 (Triangle inequality). For random variables {X, Y}:

(F.12)

Proof. Expanding the left-hand side under the square root gives

(F.13)

where the Cauchy–Schwarz inequality has been applied to both terms on the right-hand side. Dividing both sides by completes the proof.

The triangle inequality is a special case of Minkowski's inequality, which we state without proof.

Theorem F.4 (Minkowski's inequality). For random variables {X, Y} and :

(F.14)

Another form of the triangle inequality is obtained from (F.14) with p = 1:

(F.15)