On quasi-Markov random fields

In this paper we define the quasi-Markov property and give a complete characterization of a stationary Gaussian quasi-Markov process. We show that the only stationary Gaussian quasi-Markov process on the whole real line is a Markov process, but there exists a non-Markovian singular stationary Gaussian quasi-Markov process on the set of all the integers. The Markov property implies the quasi-Markov property, and a sufficient condition for a stationary quasi-Markov process to be Markov is given. Following Rozanov we define a generalized notion of boundary, called an L-boundary, and the L-Markov property, and we give a discussion of the uniqueness of a probability distribution corresponding to a family of conditional distributions for the stationary Gaussian L-Markov case. A characterization of the stationary Gaussian L-field is given in terms of the spectral distribution as well as a discussion of the relationship between L-Markov and L-field. We study the existence of a properly nonsingular-stationary Gaussian L-field and show that it can exist only in three or higher dimensions. We also discuss the technique of the best linear interpolation in mean-square and show how this technique can be used in studying the above problems.