Orthogonality_note

Orthogonality_note - Stat/219 Math 136 - Stochastic...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stat/219 Math 136 - Stochastic Processes Notes on Section 4.1.2 1 Independence, Uncorrelated-ness, and Something in Between Suppose ( X,Y ) is a random vector defined on some probability space (Ω , F , IP) with IE( X 2 ) < ∞ , IE( Y 2 ) < ∞ . The following implications are true. IE( X | Y ) = IE( X ) ր ց X and Y are independent X and Y are uncorrelated (1 A ) ց ր IE( Y | X ) = IE( Y ) For a proof of the first statements on the left above, assume X and Y are independent. Then the constant IE( X ) is measurable with respect to σ ( Y ). Since X and σ ( Y ) are independent, for any G ∈ σ ( Y ) we have integraldisplay G IE( X ) d IP = IE( X )IE( I G ) = IE( XI G ) by independence = integraldisplay G Xd IP Thus IE( X | Y ) = IE( X ). Similarly we have IE( Y | X ) = IE( Y ). Next we prove the statements on the right side of (1A). Suppose that IE( Y | X ) = IE( Y ). Using the tower property and then taking out what is known, we have IE( XY ) = IE(IE( XY | X )) = IE( X (IE( Y | X )) = IE( X )IE( Y ). Thus X and Y are uncorrelated. Similarly, IE( X | Y ) = IE( X ) implies that X and Y are uncorrelated. However, the converse implications are not true in general. Counterexample 1: X and Y uncorrelated does not imply IE( X | Y ) = IE( X ) Let Ω = {− 1 , , 1 } with P ( { ω } ) = 1 / 3 for each ω ∈ Ω. Let Y ( ω ) = ω and X ( ω ) = I { } ( ω ). Then XY = 0 so IE( XY ) = 0. Also, IE( Y ) = 0, and so IE( XY ) = 0 = IE( X )IE( Y ); that is, X and Y are uncorrelated. However, since X is measurable with respect to σ ( Y ) we have IE( X | Y ) = X which is never equal to IE( X ) = 1 / 3....
View Full Document

This note was uploaded on 11/08/2009 for the course STAT 219 taught by Professor -2 during the Fall '08 term at Stanford.

Page1 / 3

Orthogonality_note - Stat/219 Math 136 - Stochastic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online