Inequalities and Limit Theorems

The Cauchy-Schwarz inequality is one of the most famous inequalities in all of mathematics. In probability, it takes the following form.

Theorem 10.1.1 (Cauchy-Schwarz). For any r.v.s X and Y with finite variances,

|E(XY )| ≤ √ E(X2)E(Y 2).