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

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4.2. Linear estimation

The fundamental space that we define below has already been introduced in Chapter 3 but in a different context.

DEFINITION.– Space up to instant K − 1 is called linear space of observation and the vector space of the linear combinations of r.v. 1, Y1, …, Yk−1 is denoted images (or images), i.e.:

images

Since r.v. 1, Y1, …, Yk−1L2(dP) , images is a vector subspace (closed, as the number of r.v. is finite) of L2(dP).

We can also say that images is a Hilbert subspace of L2 (dP).

We are focusing here on the problem stated in the preceding section but with a simplified hypothesis: g is linear, which means that the envisaged estimators Z of XK are of the form:

images and thus belong to images.

The problem presents itself as: find the r.v., denoted , which renders minimum mapping:

(i.e., find which render minimum: ...

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