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

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.