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

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2.5. The existence of Gaussian vectors

NOTATION.– uT = (u1,…, un), xT = (x1,…, xn) and mT = (m1,…, mn).

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

images

PROPOSITION.– Given a vector mT = (m1,…, mm) and a matrix Γ ∈ M(n, n), which is symmetric and semi-defined positive, there is a unique probability PX on imagesn, of the Fourier transform:

images

In addition:

1) if Γ is invertible, PX admits on imagesn the density:

images

2) if Γ is non-invertible (of rank r < n) the r.v. X1m1,…, Xnmn are linearly dependent. We can still say that ωX (ω) − m a.s. takes its values on a hyperplane (Π) of imagesn or that the probability PX loads a hyperplane (Π) does not admit a density function on n.

DEMONSTRATION.–

1) Let us ...

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