We shall introduce the definition of a k-mode Gaussian state and a chain of such states which determine a C* probability space. We present examples of such states exhibiting properties like exchangeability and stationarity. Stationary chains are determined by block Toeplitz matrices. Using the Kac-Murdoch-Szego theorems on asymptotic spectral distributions of Toeplitz matrices we compute the entropy rates of some of these chains. This leaves many natural problems open.

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