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On the correlation structure of unilateral AR processes on the plane

Part of: Numerical approximation and computational geometry Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

F. Champagnat  and
J. Idier
F. Champagnat*
Affiliation:
Office National d'études et de Recherches Aérospatiales
J. Idier*
Affiliation:
Laboratoire des Signaux et Systèmes
*
∗ Postal address: ONERA, DTIM/TI, 29 av. de la division Leclerc, BP 72-92322 Châtillon Cedex, France.
∗∗ Postal address: Laboratoire des Signaux et Systèmes, SUPÉLEC, Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France. Email address: idier@lss.supelec.fr
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Abstract

In, Tory and Pickard show that a simple subclass of unilateral AR processes identifies with Gaussian Pickard random fields on Z 2. First, we extend this result to the whole class of unilateral AR processes, by showing that they all satisfy a Pickard-type property, under which correlation matching and maximum entropy properties are assessed. Then, it is established that the Pickard property provides the ‘missing’ equations that complement the two-dimensional Yule-Walker equations, in the sense that the conjunction defines a one-to-one mapping between the set of AR parameters and a set of correlations. It also implies Markov chain conditions that allow exact evaluation of the likelihood and an exact sampling scheme on finite lattices.

Keywords

Quarter-plane ARMarkov random fieldsunilateral Gaussian fieldsPickard random fields2-D maximum entropy

MSC classification

Primary: 60G10: Stationary processes 60G25: Prediction theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 65D15: Algorithms for functional approximation

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 408 - 425
Copyright
Copyright © Applied Probability Trust 2000 

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