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 :
Another form of the triangle inequality is obtained from (F.14) with p = 1:
(F.15)
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