Discrete Stochastic Processes and Optimal Filtering

2.5. The existence of Gaussian vectors

NOTATION.– u^T = (u₁,…, u_n), x^T = (x₁,…, x_n) and m^T = (m₁,…, m_n).

We are interested here in the existence of Gaussian vectors, that is to say the existence of laws of probability on ⁿ having Fourier transforms of the form:

PROPOSITION.– Given a vector m^T = (m₁,…, m_m) and a matrix Γ ∈ M(n, n), which is symmetric and semi-defined positive, there is a unique probability P_X on ⁿ, of the Fourier transform:

In addition:

1) if Γ is invertible, P_X admits on ⁿ the density:

2) if Γ is non-invertible (of rank r < n) the r.v. X₁ − m₁,…, X_n − m_n are linearly dependent. We can still say that ω → X (ω) − m a.s. takes its values on a hyperplane (Π) of ⁿ or that the probability P_X loads a hyperplane (Π) does not admit a density function on ⁿ.

DEMONSTRATION.–

1) Let us ...