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A curious binary lattice process

Published online by Cambridge University Press:  14 July 2016

D. K. Pickard
D. K. Pickard*
Affiliation:
The Australian National University
*
*Now at Harvard University.
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Abstract

A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field on Z × Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.

Keywords

GIBBS ENSEMBLESLATTICE PROCESSES0 − 1 MARKOV RANDOM FIELDSGROWTH MODELSMARKOV CHAIN CONSTRUCTIONGEOMETRIC CORRELATIONSDIRECT SIMULATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 717 - 731
Copyright
Copyright © Applied Probability Trust 1977 

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References

Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192–236.Google Scholar
Galbraith, R. and Walley, D. (1976) On a two dimensional binary process. J. Appl. Prob. 13, 548–557.Google Scholar
Minlos, R. A. (1967) Limiting Gibbs distributions. Funct. Anal. Appl. 1, 140–150.Google Scholar
Moran, P. A. P. (1968) An Introduction to Probability Theory. Clarendon Press, Oxford.Google Scholar
Moussouris, J. (1974) Gibbs and Markov random systems with constraints. J. Statist. Phys. 10, 11–33.Google Scholar
Pickard, D. K. (1977) Statistical Inference on Lattices. , The Australian National University.Google Scholar
Pickard, D. K. (1978) Unilateral Ising models. In Spatial Patterns and Processes, Suppl. Adv. Appl. Prob. 10, 58–64.Google Scholar
Preston, C. J. (1974) Gibbs States on Countable Sets. Cambridge University Press.Google Scholar
Verhagen, A. M. W. (1977) A three parameter isotropic distribution of atoms and the hard-core square lattice gas. To appear.Google Scholar
Welberry, T. R. and Galbraith, R. (1973) A two-dimensional model of crystal growth disorder. J. Appl. Cryst. 6, 87–96.Google Scholar
Welberry, T. R. and Galbraith, R. (1975) The effect of non-linearity on a two-dimensional model of crystal growth disorder. J. Appl. Cryst. 8, 636–644.Google Scholar
Welberry, T. R. (1977) Solution of crystal growth disorder models by imposition of symmetry. Proc. R. Soc. A (To appear).Google Scholar