**2.5. The existence of Gaussian vectors**

NOTATION.– *u*^{T} = (*u*_{1},…, *u*_{n}), *x*^{T} = (*x*_{1},…, *x*_{n}) and *m*^{T} = (*m*_{1},…, *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} = (*m*_{1},…, *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. *X*_{1} − *m*_{1},…, *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 ...