## The Annals of Applied Probability

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

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1177004839

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aoap/1177004839

**Mathematical Reviews number (MathSciNet)**

MR1325052

**Zentralblatt MATH identifier**

0822.60045

**Subjects**

Primary: 60G60: Random fields

Secondary: 60G25: Prediction theory [See also 62M20]

**Keywords**

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

#### Citation

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. http://projecteuclid.org/euclid.aoap/1177004839.