We are seeking to improve the result by considering as estimation of XK not only the linear functions of r.v. Y1, …, YK−1 but the general functions
PROPOSITION.– The family of r.v. Borel functions; is a closed vector subspace of L2.
Let us note again = Hilbert space equipped with a scalar product: .
Furthermore, fY (y1, …, yK − 1) designating the density of the vector Y = (Y1, …, YK − 1), in order to simplify its expression let us state:
and let us introduce the new Hilbert space:
This is equipped with the scalar product: ∀g1, g2 ∈ L2 (dμ)
Thus finally the linear mapping:
We notice that ψ conserves the scalar product (and the ...