Porwards and Backwards Models for Finite-state Markov Processes

A commonly considered problem is that of spectral factorization: given the power spectrum of a second-order stationary process, find a time-invariant linear system such that excitation of the linear system by white noise results in the output process having the prescribed spectrum. In this paper, we are still concerned with the generation of a rational power spectrum, but now via a different procedure. We conceive of a finite-state Markov process as generating the power spectrum, following ideas in for example 131. Our main task in this paper is to investigate the applicability of several ideas developed in [lo] on the time-reversal of spectrum models. More precisely, in [lo] it was shown that given a fmite-dimensional lmear system driven by white noise running, as usual, forwards in time, there exists an associated system running backwards in time generating the same covariance. The initial state in the forwards model is independent of future noise, while the terminal state in the backwards model is independent of past noise. The first question that then arises in connection with finite-state Markov models is whether, given a fonvard-time model generating a certain power spectrum, there exists an associated backward-time finite-state Markov model, generating the same