  We explore conditions on the covariance matrices of a consistent chain of
mean zero finite mode Gaussian states in order that the chain may be
exchangeable or stationary. For an exchangeable chain our conditions are
necessary and sufficient. Every stationary Gaussian chain admits an asymptotic
entropy rate. Whereas an exchangeable chain admits a simple expression for its
entropy rate, in our examples of stationary chains the same admits an integral
formula based on the asymptotic eigenvalue distribution for Toeplitz matrices.
An example of a stationary entangled Gaussian chain is given.
