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A regularity criterion for lexicographical prediction of multivariate wide-sense stationary processes on \(\mathbb{Z}^ 2\) with non-full-rank spectral densities. (English) Zbl 0772.60028

The problem of regularity of multivariate wide-sense stationary sequences for the full rank, as well as the nonfull rank case is well studied, and its spectral characterization is well-known, see the review paper by P. Masani [Multivariate analysis, Proc. Int. Sympos. Dayton 1965, 351–382 (1966; Zbl 0216.46706)]. This characterization with the emphasize placed on the analyticity of range functions is contained in the work of H. Helson and D. Lowdenslager [Acta. Math. 106, 175–213 (1961; Zbl 0102.34802)]. The most important result of the paper under review is Theorem 3.2. It provides a necessary and sufficient spectral characterization for the regularity, with respect to the family of “half spaces” introduced by H. Helson and D. Lowdenslager [Acta. Math. 99, 165–202 (1958; Zbl 0082.28201)], of multivariate two-parameter random fields with nonfull rank spectral density, and thereby providing an extension of Helson and Lowdenslager’s work for the one-parameter case. The paper under review treats the two-discrete parameter case only and contains several other important and interesting results.

MSC:

60G25 Prediction theory (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
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References:

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