A well-known theorem in probability states: Let x be an Nx1 Gaussian vector with mean vector m and covariance matrix Cx .If we linearly transform x as y=Px using a square NxN transform matrix P, then y is also Gaussian with mean vector Pm and covariance matrix Cy=PCxPT.
This theorem has an elegant proof using the multivariate Gaussian probability density function (pdf).
It is also known that the above theorem is also valid for non-square (NxM) transform matrices, P.
Does anyone know any reference book containing the proof for the more general case of non-square linear transformations?
Thank you very much Dr. Dijkstra. Your proof and your references were very helpful and informative.
I don't think it is valid for general P. Example: consider a 2-variate normal distribution X=[X1;X2] and P = [1, 0; 1, 0]. Hence, PX = [X1; X1] which is no longer a 2-variate normal distribution even though each element is normally distributed. This can be seen by the following definition.
Definition:A mx1 random vector X is said to have a m-variate normal distribution if for every vector v consisting of m real constants the distribution v'X is univariate normal (see e.g. "Aspects of Multivariate Statistical Theory" by Murihead)
So, for our example, take v = [1;-1] which gives v'(PX) = 0.
I think what you need is an additional rank condition for your transform matrix. Then you could refer to the definition to confirm that PX is a m-variate normal without having to compute characteristic functions. Its moments readily follow.
Dear Daniel,
In my question, I was concerned (albeit implicitly) with P matrices which are not rank-deficient. For those cases, the theorem works perfectly. Although, as I understand from Dr. Dikstra's answer above, the related literature somehow handles the contrived degenerate cases as well. Thank you all for your insightful answers...
Olcay
ok. but then it just is by definition normal, no need to use characteristic functions to prove anything
(Actually, Jet Propulsion Laboratory)
Try this approach.
Definition: the scalar random variable x is Gaussian if either x has a Gaussian density or x is a constant.
Definition: the random N-vector x is Gaussian if, for any constant N-vector a, a'x is a Gaussian random variable.
As a consequence, each component x(i) of a Gaussian random vector x is a Gaussian random variable, and so Ex and cov x := E(x-Ex)(x-Ex)' exist.
Let x be a Gaussian vector. Let y = Px for some M x N matrix P. For any M-vector b, b'y = (b'P)x, which by definition is a Gaussian random variable. Therefore y is a Gaussian M-vector, and
Ey = P(Ex),
cov y = E(y - Ey)(y - Ey)' = PE(x - Ex)(x - Ex)'P' = P(cov x)P'
See Rosenblatt, Random Processes, Oxford U. Press.
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