The Annals of Applied Probability

A Wold-Like Decomposition of Two-Dimensional Discrete Homogeneous Random Fields

Joseph M. Francos, A. Zvi Meiri, and Boaz Porat

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Imposing a total order on a regular two-dimensional discrete random field induces an orthogonal decomposition of the random field into two components: a purely indeterministic field and a deterministic field. The deterministic component is further orthogonally decomposed into a half-plane deterministic field and a countable number of mutually orthogonal evanescent fields. Each of the evanescent fields is generated by the column-to-column innovations of the deterministic field with respect to a different nonsymmetrical-half-plane total-ordering definition. The half-plane deterministic field has no innovations, nor column-to-column innovations, with respect to any nonsymmetrical-half-plane total-ordering definition. This decomposition results in a corresponding decomposition of the spectral measure of the regular random field into a countable sum of mutually singular spectral measures.

Article information

Ann. Appl. Probab., Volume 5, Number 1 (1995), 248-260.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G60: Random fields
Secondary: 60G25: Prediction theory [See also 62M20]

Two-dimensional random fields two-dimensional Wold decomposition purely indeterministic random fields deterministic random fields evanescent random fields two-dimensional spectral measures


Francos, Joseph M.; Meiri, A. Zvi; Porat, Boaz. A Wold-Like Decomposition of Two-Dimensional Discrete Homogeneous Random Fields. Ann. Appl. Probab. 5 (1995), no. 1, 248--260. doi:10.1214/aoap/1177004839.

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