Hilbert space representations of general discrete time stochastic processes (Q1060483)

We show that under mild conditions the joint densities \(p_{X_ 1,...,X_ n}(x_ 1,...,x_ n)\) of the general discrete time stochastic process \(X_ n\) on \({\mathcal H}\) can be computed via \(p_{X_ 1,...,X_ n}(x_ 1,...,x_ n)=\| \phi T(x_ 1)\cdot \cdot \cdot T(x_ n)\|^ 2\) where \(\phi\) is in the Hilbert space \({\mathcal H}\), and T(x), \(x\in {\mathcal H}\), are linear operators on \({\mathcal H}\). We then show how the central limit theorem can easily be derived from such representations.