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

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1.5. Linear independence of vectors of L2 (dP)

DEFINITION.– We say that n vectors X1, …, Xn of L2 (dP) are linearly independent if images a.s. images (here 0 is the zero vector of L2 (dP)).

DEFINITION.– We say that the n vectors X1, …, X2 of L2 (dP) are linearly dependent if ∃ λ1, …,λn are not all zero and ∃ an event A of positive probability such that λ1X1 (ω) +… + λnXn (ω) = 0 ∀ωA.

In particular: X1, …, Xn will be linearly dependent if ∃ λ1, …,λn are not all zero such that λ1 X1 +… + λn Xn = 0 a.s.

Examples: given the three measurable mappings:

images

defined by:

images

image

Figure 1.6. Three random variables

The three mappings are evidently measurable and belong to L2(), so there are 3 vectors of L2().

There 3 vectors are linearly dependent on A = [0,1[ of probability measurement image:

images

Covariance matrix and linear ...

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