Discrete Stochastic Processes and Optimal Filtering

1.5. Linear independence of vectors of L² (dP)

DEFINITION.– We say that n vectors X₁, …, X_n of L² (dP) are linearly independent if a.s. (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, …,λ_n are not all zero and ∃ an event A of positive probability such that λ₁X₁ (ω) +… + λ_nX_n (ω) = 0 ∀ω∈ A.

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

Examples: given the three measurable mappings:

defined by:

Figure 1.6. Three random variables

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

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

Covariance matrix and linear ...