Lecture6

Lecture6 - Lecture 6 : Distributions Theorem 6.0.1 (Holders...

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Unformatted text preview: Lecture 6 : Distributions Theorem 6.0.1 (Holders Inequality) If p,q [1 , ] with 1 /p + 1 /q = 1 then E ( | XY | ) || X || p || Y || q (6.1) Here || X || r = ( E ( | X | r )) 1 /r for x [1 , ) ; and || X || = inf { M : P ( | X | > M ) = 0 } . Proof: See the proof of (5.2) in the Appendix of Durrett. Example 6.0.2 If | Y | b then E ( | XY | ) b E ( | X | ) Theorem 6.0.3 (Cauchy-Schwarz Inequality) The special case p = q = 2 is the Cauchy-Schwarz inequality. E ( | XY | ) ( E ( X 2 ) E ( Y 2 )) 1 / 2 (6.2) Proof: Apply Holders inequality for p = q = 2. Example 6.0.4 For X take Y = 1 X . Then we get E ( X ) = E ( | XY | ) E ( X 2 ) 1 / 2 E ( Y 2 ) 1 / 2 = E ( X 2 ) 1 / 2 E ( Y ) 1 / 2 = E ( X 2 ) 1 / 2 P ( X > 0) 1 / 2 so squaring both sides we get P ( X > 0) E ( X ) 2 E ( X 2 ) . Theorem 6.0.5 (Minkowskis Inequality (Triangle inequality for L p )) || X + Y || p || X || p + || Y || p 6-1 Lecture 6: Distributions 6-2 6.1 Probability distribution on the real line6....
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Lecture6 - Lecture 6 : Distributions Theorem 6.0.1 (Holders...

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