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 n having Fourier transforms of the form:
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 n, of the Fourier transform:
1) if Γ is invertible, PX admits on n the density:
2) if Γ is non-invertible (of rank r < n) the r.v. X1 − m1,…, Xn − mn 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 PX loads a hyperplane (Π) does not admit a density function on n.
1) Let us ...