Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-04-30T10:57:38.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Necessary conditions for Markovian processes on a lattice

Published online by Cambridge University Press:  14 July 2016

P. A. P. Moran
P. A. P. Moran*
Affiliation:
The Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

Stationary processes which are defined on the points of a square lattice and are Markovian in various senses are considered. It is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and that given this form Gaussian processes of this kind are easily constructed. Certain non-Gaussian processes satisfying the various Markovian properties are also constructed and the difference from nearest-neighbour systems emphasized. It is conjectured, but not proved, that the assumption of linearity of regression also implies Gaussianity.

Keywords

MARKOV PROCESSES ON LATTICESSPECTRAL THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 605 - 612
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

Bartlett, M. S. (1971) Physical nearest neighbour systems and non-linear time series. J. Appl. Prob. 8, 222–232.10.2307/3211892Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Hammersley, J. M. and Clifford, P. (1972) Markov fields on finite graphs and lattices. To appear.Google Scholar
Levy, P. (1948) Processus stochastiques et movement Brownien. Gautier-Villars, Paris.Google Scholar
Moran, P. A. P. (1973) A Gaussian Markovian process on a square lattice. J. Appl. Prob. 10, 54–62.Google Scholar
Rosanov, Yu. A. (1967) On Gaussian fields with given conditional distributions. Theor. Probability Appl. 12, 381–391.Google Scholar
Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142–154.10.1080/00029890.1971.11992710Google Scholar
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–449.10.1093/biomet/41.3-4.434Google Scholar
Zygmund, A. (1968) Trigonometric Series. Cambridge University Press.Google Scholar