Show that if ξ is a random variable in L 2 ( F ) and G is a σ -field contained in F, then E ( ξ | G ) is the orthogonal projection of ξ onto the subspace L 2 ( G ) in L 2 ( F ) consisting of G-measurable random variables.

Question

Show that if   ξ is a random variable in       L    2    (  F  ) and G is a   σ -field contained in F, then   E  (  ξ      |    G  ) is the orthogonal projection of   ξ onto the subspace       L    2    (  G  ) in       L    2    (  F  ) consisting of G-measurable random variables.

enreciarpv · Accepted Answer

Step 1Recall that for a Hilbert space H and its closed, convex, non-empty subset C the projection of a point x on to C is the unique point   y  ∈  C such that   ‖  x  −  y  ‖  =      inf          z      ∈      C        ‖  x  −  z  ‖ .I will write PG for the orthogonal projection on to       L    2    (  G  ) . We want to show       P    G    ξ  =      E    [  ξ      |    G  ] almost surely.Let   Y  =      P    G    ξ and take   Z  ∈      L    2    (  G  ) . Consider   W  =  Y  +  a  Z  ∈      L    2    (  G  ) for   a  ∈      R   . It follows from the definition of the projection that  0  ≤  ‖  ξ  −  W      ‖                  L        2            (      F      )        2    −  ‖  ξ  −  Y      ‖                  L        2            (      F      )        2    =      a    2        E    [      Z    2    ]  −  2  a      E    [  (  ξ  −  Y  )  Z  ]The right hand side is a quadratic in a which is non-negative for all a iff       E    [  (  ξ  −  Y  )  Z  ]  =  0 so   ξ  −  Y  ⊥      L    2    (  G  ) . (This is a special case of the more general fact that if P is an orthogonal projection on to a subspace U of a Hilbert space H then for   x  ∈  H ,   x  −  P  x  ⊥  U )By considering Z of the form       1    A   for   A  ∈  G , it immediately follows that Y is a version of       E    [  ξ      |    G  ] and we are done.

Show that if xi is a random variable in L^2(F) and G is a sigma -field contained in F, then E(xi|G) is the orthogonal projection of xi onto the subspace L^2(G) in L^2(F) consisting of G-measurable random variables.

Answered question

Answer & Explanation

New Questions in Upper Level Math