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On the correlation structure of unilateral AR processes on the plane

Part of: Numerical approximation and computational geometry Special processes Stochastic 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 Z2. 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
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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