On quasi-Markov random fields

S. C. Chay

Journal of Multivariate Analysis, 1972, vol. 2, issue 1, 14-76

Abstract: 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.

Keywords: Random; field; stochastic; process; Gaussian; random; field; stationary; process; quasi-Markov; property; Markov; property; spectral; representation; linear; interpolation; in; mean-square; uniqueness; of; probability; distribution; characterization; of; stationary; Gaussian; L-field; regular; field; singular; field; existence; of; nonsingular; random; field (search for similar items in EconPapers)
Date: 1972
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Citations: View citations in EconPapers (2)

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