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, *Y*_{1}, …, *Y*_{k−1} is denoted (or ), i.e.:

Since r.v. 1, *Y*_{1}, …, *Y*_{k−1} ∈ *L*^{2}(*dP*) , is a vector subspace (closed, as the number of r.v. is finite) of *L*^{2}(*dP*).

We can also say that is a Hilbert subspace of *L*^{2} (*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 *X _{K}* are of the form:

and thus belong to .

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

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

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