Spectral Theory of Circuit Processes

Abstract

Spectral theory of Markov processes was developed by D.G. Kendall (1958, 1959a, b) and W. Feller (1966a). The present chapter relies on Kendall’s Fourier representation for transition-probability matrices and for transitionmatrix functions defining discrete and continuous parameter Markov processes, respectively. A specialization of the spectral theory to circuit Markov processes is particularly motivated by the essential rôle of the circuit-weights when they decompose the finite-dimensional distributions. For this reason we shall be consequently interested in the spectral representation of the circuit-weights alone. This approach is due to S. Kalpazidou (1992a, b).