  We study the properties of stationary G-chains in terms of their generating
functions. In particular, we prove an analogue of the Szeg\H{o} limit theorem
for symplectic eigenvalues, derive an expression for the entropy rate of
stationary quantum Gaussian processes, and study the distribution of symplectic
eigenvalues of truncated block Toeplitz matrices. We also introduce a concept
of symplectic numerical range, analogous to that of numerical range, and study
some of its basic properties, mainly in the context of block Toeplitz
operators.
