  Stochastic models share many characteristics with generic parametric models.
In some ways they can be regarded as a special case. But for stochastic models
there is a notion of weak distribution or generalised random variable, and the
same arguments can be used to analyse parametric models. Such models in vector
spaces are connected to a linear map, and in infinite dimensional spaces are a
true gener- alisation. Reproducing kernel Hilbert space and affine- / linear-
representations in terms of tensor products are directly related to this linear
operator. This linear map leads to a generalised correlation operator, and
representations are connected with factorisations of the correlation operator.
The fitting counterpart in the stochastic domain to make this point of view as
simple as possible are algebras of random variables with a distinguished linear
functional, the state, which is interpreted as expectation. The connections of
factorisations of the generalised correlation to the spectral decomposition, as
well as the associated Karhunen-Lo\`eve- or proper orthogonal decomposition
will be sketched. The purpose of this short note is to show the common
theoretical background and pull some lose ends together.
