Hostname: page-component-6bb9c88b65-6scc2 Total loading time: 0 Render date: 2025-07-22T00:44:47.335Z Has data issue: false hasContentIssue false
  • English
  • Français

Unilateral Gaussian fields

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

E. M. Tory  and
D. K. Pickard
E. M. Tory*
Affiliation:
Mount Allison University
D. K. Pickard*
Affiliation:
Harvard University
*
∗Postal address: Department of Mathematics and Computer Science, Mount Allison University, Sackville, N.B., EOA 3C0, Canada.
∗∗D. K. Pickard died on 28 July 1986. This joint paper was later completed by E. M. Tory.
Get access Rights & Permissions [Opens in a new window]

Abstract

The necessary and sufficient condition for unilateral characterization of Gaussian Markov fields and the Besag-Moran positivity condition for second-order autonormal bilateral models define the same tetrahedral domain of achievable regression parameters. A bijective function maps this domain to a different tetrahedral domain of parameters in the Pickard model. These two domains are identical to the corresponding ones in the Welberry-Carroll model. We obtain series solutions for correlation coefficients and study their limits near the boundaries of the first domain.

Keywords

MARKOV FIELDSISING MODELSCRYSTAL GROWTH-DISORDER MODELSUNILATERAL PROCESSESCORRELATIONS OF EMBEDDED MARKOV CHAINS

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 95 - 112
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Article purchase

Temporarily unavailable

Footnotes

Research supported in part by the Natural Sciences and Engineering Research Council of Canada.

References

Bartlett, M. S. (1967) Inference and stochastic processes. J. R. Statist. Soc. A 130, 457–477.Google Scholar
Besag, J. E. (1972) On the correlation structure of some two-dimensional stationary processes. Biometrika 59, 43–48.Google Scholar
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192–225.Google Scholar
Besag, J. (1981) On a system of two-dimensional recurrence equations. J. R. Statist. Soc. B 43, 302–309.Google Scholar
Besag, J. E. and Moran, P. A. P. (1975) On the estimation and testing of spatial interaction in Gaussian lattice processes. Biometrika 62, 555–562.Google Scholar
Enting, I. G. (1978) Crystal growth models and Ising models III. Zero field susceptibilities and correlations. J. Phys. A 11, 555–562.Google Scholar
Feller, W. (1968) Introduction to Probability Theory , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Galbraith, R. F. and Walley, D. (1976) On a two-dimensional binary process. J. Appl. Prob. 13, 548–557.Google Scholar
Galbraith, R. F. and Walley, D. (1982) Further properties for unilateral binary processes. J. Appl. Prob. 19, 332–343.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
Moran, P. A. P. (1973) A Gaussian Markovian process on a square lattice. J. Appl. Prob. 10, 54–62.Google Scholar
Pickard, D. K. (1977) A curious binary lattice process. J. Appl. Prob. 14, 717–731.Google Scholar
Pickard, D. K. (1978) Unilateral Ising models. Suppl. Adv. Appl. Prob. 10, 58–64.Google Scholar
Pickard, D. K. (1980) Unilateral Markov fields. Adv. Appl. Prob. 12, 655–671.Google Scholar
Speed, T. P. (1978) Relations between models for spatial data, contingency tables and Markov fields on graphs. Suppl. Adv. Appl. Prob. 10, 111–122.Google Scholar
Verhagen, A. M. W. (1977) A three parameter isotropic distribution of atoms and the hard-core square lattice gas. J. Chem. Phys. 67, 5060–5065.Google Scholar
Welberry, T. R. (1977) Solution of crystal growth disorder models by imposition of symmetry. Proc. R. Soc. London A 353, 363–376.Google Scholar
Welberry, T. R. (1985) Diffuse X-ray scattering and models of disorder. Rep. Prog. Phys. 48, 1543–1593.Google Scholar
Welberry, T. R. and Carroll, C. E. (1982) Gaussian growth-disorder models and optical transform methods. Acta Cryst. A 38, 761–772.Google Scholar
Welberry, T. R. and Carroll, C. E. (1983) Further properties of a Gaussian model of disorder. Acta Cryst. A 39, 233–245.Google Scholar
Welberry, T. R., Miller, G. H., and Carroll, C. E. (1980) Paracrystals and growth-disorder models. Acta Cryst. A 36, 921–929.Google Scholar
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–449.Google Scholar