Conditional expectation and Least Squares Regression

[piolo]

Hello everybody,

I have two questions on conditional expectation w.r.t (Polynomial) OLS:
Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how we can prove the following statements are true/false:

(feel free to add assumptions)

1. OLS( f(X_T), Vect_n{X_t} ) -> E( f(X_T) | F_t ), when n-> \infty
2. Norm_L2{ E( f(X_T) | F_T ) - OLS( (X_T), Vect_n{X_T} ) } >= Norm_L2{ E( f(X_T) | F_t ) - OLS( (X_T), Vect_n{X_t} ) }

For the first one, suppose X_t is Markov + Stone-Weierstrass + projection, we may have something interesting. But for the second one, I don't have any idea...

Any help? Thx.

 

[piolo]

Can anyone tell me if there is some (or no) relationship between the projection of f(X_T) on Vector space{X_t^0, X_t^1, ..., X_t^m} and the projection of f(X_T) on Vector space{X_T^0, X_T^1, ..., X_T^m}? Where X_t is a markov process.(or any other appropriate process).