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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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4.3. Best estimate – conditional expectation

We are seeking to improve the result by considering as estimation of XK not only the linear functions images of r.v. Y1, …, YK−1 but the general functions images

PROPOSITION.– The family of r.v. images Borel functions; images is a closed vector subspace of L2.

DEMONSTRATION.–

Let us note again images = Hilbert space equipped with a scalar product: images.

Furthermore, fY (y1, …, yK − 1) designating the density of the vector Y = (Y1, …, YK − 1), in order to simplify its expression let us state:

images

and let us introduce the new Hilbert space:

images

This is equipped with the scalar product: ∀g1, g2L2 ()

Thus finally the linear mapping:

We notice that ψ conserves the scalar product (and the ...

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