Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T10:06:01.716Z Has data issue: false hasContentIssue false

A Gaussian Markovian process on a square lattice

Published online by Cambridge University Press:  14 July 2016

P. A. P. Moran*
Affiliation:
The Australian National University

Abstract

A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abramowitz, M. and Stegum, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Bartlett, M. S. (1971) Physical nearest neighbour systems and non-linear time series. J. Appl. Prob. 8, 222232.Google Scholar
[3] Hammersley, J. M. and Clifford, P. Markov fields on finite graphs and lattices. To appear.Google Scholar
[4] Katsura, S. and Inawashiro, S. (1971) Lattice Green's function for the rectangular and square lattices at arbitrary points. J. Math. and Phys. 12, 16221630.Google Scholar
[5] Levy, P. (1948) Processus Stochastiques et Movement Brownien. Gautier Villars, Paris.Google Scholar
[6] Montroll, E. W. (1964) Random walks on lattices. Proceedings of Symposia in Applied Mathematics 16, 193220.Google Scholar
[7] Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142154.Google Scholar
[8] Titchmarsh, E. C. (1937) Introduction to the Theory of Fourier Integrals. Oxford.Google Scholar
[9] Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434449.Google Scholar
[10] Wong, E. (1969) Homogeneous Gauss-Markov random fields. Ann. Math. Statist. 40, 16251634.Google Scholar
[11] Rosanov, Yu. A. (1967) On the Gaussian homogeneous fields with given conditional distributions. Theor. Probability Appl. 12, 381391.Google Scholar