Gaussian measures dichotomy theorem

Let consist of functions x(t), , and let be the -algebra spanned by the cylinder sets. A PD P on is called a Gaussian measure if all of its finite dimensional distributions are Gaussian. Here, for , is the image of P under the mapping .

The dichotomy theorem says that two Gaussian measures P and Q on are either equivalent (mutually absolutely continuous) or orthogonal (singular detection is possible: there exists with P(A)=1, Q(A)=0).

The first proof of this important result was obtained via IT (Hájek 1958), by showing that for Gaussian measures, implies orthogonality. Of course, implies equivalence, even if P and Q are non-Gaussian.

Sketch of Hájek's proof:

1. 
   (an easy consequence of eq. (1.2)).
2. I-divergence is invariant under one-to-one transformations, this permits us to reduce calculation of to the ``easy case'' when is (k-dimensional) standard Gaussian and also is of product form.
3. In the ``easy case'' above, direct calculation shows that if is ``large'' then and are ``almost concentrated on disjoint sets''.