Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T20:43:37.727Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for conditionally centred random fields with an application to Markov fields

Part of: Stochastic processes Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Francis Comets  and
Martin Janžura
Francis Comets*
Affiliation:
Université Paris 7
Martin Janžura*
Affiliation:
Academy of Sciences, Prague
*
∗Postal address: Université Paris 7 – Denis Diderot, Mathématiques, case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France. Email address: comets@math.jussieu.fr.
∗∗Postal address: Institute of Information Theory and Automation, Academy of Sciences, Pod vodárenskou věží 4, CZ – 182 08 Praha, Czech Republic.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

Keywords

Central limit theoremMarkov random fieldsconditional centring

MSC classification

Primary: 60G60: Random fields
Secondary: 60F05: Central limit and other weak theorems 62M40: Random fields; image analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 608 - 621
Copyright
Copyright © Applied Probability Trust 1998 

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.)

Footnotes

Unité de Recherche Associée CNRS 1321 ‘Statistique et Modèles Aléatoires’.

Supported by GA ČR Grant No. 202/93/0449.

References

Basawa, I. V. and Scott., D. J. (1983). Asymptotic Optimal Inference for Non-Ergodic Models. Lecture Notes in Statistics 17, Springer, New York.Google Scholar
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B, 36, 192–236.Google Scholar
Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Prob. 10, 1047–1050.CrossRefGoogle Scholar
Comets, F. (1992). On consistency of a class of estimators for exponential families of Markov random fields on a lattice. Ann. Statist. 20, 455–468.Google Scholar
Cox, J. T., and Grimmett, G. R. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Prob. 12, 514–528.Google Scholar
Dobrushin, R. L., and Nahapetian, B. S. (1974). Strong convexity of the pressure for lattice systems of classical statistical physics (in Russian). Teoret. Mat. Fiz. 20, 223–234.Google Scholar
Doukhan, P. (1994). Mixing. Lecture Notes in Statistics 85, Springer, New York.Google Scholar
Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin.Google Scholar
Georgii, H. O. (1993). Large deviations and maximum entropy principle for interacting random fields on ℤ d . Ann. Prob. 21, 1845–1874.Google Scholar
Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. De Gruyter, Berlin.Google Scholar
Götze, F., and Hipp, C. (1990). Local limit theorem for sums of finite range potentials of a Gibbsian random field. Ann. Prob. 18, 810–828.Google Scholar
Guyon, X. (1993). Champs aléatoires sur un réseau. Modélisation, statistique et applications. Masson, Paris.Google Scholar
Guyon, X. and Künsch, H. R. (1992). Asymptotic comparison of estimators in the Ising model. Lecture Notes in Statistics 74, Springer, Berlin, pp. 177–198.Google Scholar
Janžura, M., and Lachout, P. (1995). A central limit theorem for stationary random fields. Math. Meth. Statist. 4, 463–472.Google Scholar
Jensen, J. L. and Künsch, H. R. (1994). On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Statist. Math. 46, 475–486.Google Scholar
Nahapetian, B., and Petrosian, A. N. (1992). Martingale-difference Gibbs random fields and central limit theorem. Ann. Acad. Sci. Fennicae, Series A–I 17, 105–110.Google Scholar
Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128.Google Scholar
Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston.Google Scholar
Stein, Ch. (1973). A bound for the error in the normal approximation of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 583–602.Google Scholar