Theorem (Cauchy-Schwartz)
For rvs X , Y ∈ L 2 ( Ω , F , P ) X,Y\in \mathscr{L}^{2}(\Omega,\mathcal{F},\mathbb{P}) X , Y ∈ L 2 ( Ω , F , P ) E [ ∣ X Y ∣ ] ≤ E [ X 2 ] E [ Y 2 ] \mathbb{E}[|XY|]\le \sqrt{ \mathbb{E}[X^{2}] }\sqrt{ \mathbb{E}[Y^{2}] } E [ ∣ X Y ∣ ] ≤ E [ X 2 ] E [ Y 2 ]
\begin{proof} Note first that a b ≤ 1 2 ( a 2 + b 2 ) , ∀ a , b ≥ 0 ab\le \frac{1}{2}(a^{2}+b^{2}),\forall a,b\ge {0} ab ≤ 2 1 ( a 2 + b 2 ) , ∀ a , b ≥ 0 rv A , B ≥ 0 A,B\ge 0 A , B ≥ 0 E [ A B ] ≤ 1 2 ( E [ A 2 ] + E [ B 2 ] ) . \mathbb{E}[AB]\le \frac{1}{2}(\mathbb{E}[A^{2}]+\mathbb{E}[B^{2}]). E [ A B ] ≤ 2 1 ( E [ A 2 ] + E [ B 2 ]) . A = ∣ X ∣ E [ X 2 ] , B = ∣ Y ∣ E [ Y 2 ] A=\frac{|X|}{\sqrt{ \mathbb{E_{}\left[ X^{2} \right]} }},B=\frac{|Y|}{\sqrt{ \mathbb{E}[Y^{2}] }} A = E [ X 2 ] ∣ X ∣ , B = E [ Y 2 ] ∣ Y ∣ E [ ∣ X ∣ ⋅ ∣ Y ∣ ] E [ X 2 ] E [ Y 2 ] ≤ 1 2 ( 1 + 1 ) ⟺ E [ ∣ X ∣ ⋅ ∣ Y ∣ ] ≤ E [ X 2 ] E [ Y 2 ] \frac{\mathbb{E}[|X|\cdot|Y|]}{\sqrt{ \mathbb{E}[X^{2}] }\sqrt{ \mathbb{E}[Y^{2}] }}\le \frac{1}{2}(1+1)\iff \mathbb{E}[|X|\cdot|Y|]\le\sqrt{ \mathbb{E}[X^{2}] }\sqrt{ \mathbb{E}[Y^{2}] } E [ X 2 ] E [ Y 2 ] E [ ∣ X ∣ ⋅ ∣ Y ∣ ] ≤ 2 1 ( 1 + 1 ) ⟺ E [ ∣ X ∣ ⋅ ∣ Y ∣ ] ≤ E [ X 2 ] E [ Y 2 ] \end{proof}
Cor
X ∈ L 2 ⟹ X ∈ L 1 X\in \mathscr{L}^{2}\implies X\in \mathscr{L}^{1} X ∈ L 2 ⟹ X ∈ L 1
Cor
X , Y ∈ L 2 ⟹ X + Y , c + X ∈ L 2 , c ∈ R X,Y\in \mathscr{L}^{2}\implies X+Y,c+X\in \mathscr{L}^{2},c\in \mathbb{R} X , Y ∈ L 2 ⟹ X + Y , c + X ∈ L 2 , c ∈ R