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Multivariate prediction and matrix Szegő theory. (English) Zbl 1285.60038
The paper is a multivariate sequel to a survey article on Szegő’s theorem and orthogonal polynomials on the unit circle (OPUC), published by N. H. Bingham in [Probab. Surv. 9, 325–339 (2012; Zbl 1285.60038)]. Similarly to the scalar version, the author starts with the Kolmogorov isomorphism theorem, which now relates an \(l\)-dimensional complex-valued zero-mean covariance stationary stochastic process \( X = (X_n : n\in{\mathbb Z})\) to an \(l\)-dimensional process \(Y\) on the unit circle \(\mathbb T\) with orthogonal increments, and an \(l\times l\) matrix-valued probability measure \(\mu\) on \(\mathbb T\). In the multivariate setting, the Verblunsky coefficients, which have been studied by D. Damanik et al. [Surv. Approx. Theory 4, 1–85 (2008; Zbl 1193.42097)], are \(l\times l\) matrices \({\alpha}_n\) such that \(\| {\alpha}_n \| < 1\), where \(\| \cdot \|\) denotes the Euclidean norm. Their multivariate Verblunsky theorem establishes a bijection between the sequences \(\alpha = ({\alpha}_n)^{\infty}_{n=1}\) of the Verblunsky coefficients and the non-trivial \(l\times l\) matrix-valued probability measures \(\mu\) on \(\mathbb T\).
The rest of the paper reports on multivariate extensions of various conditions involving prediction theory and Szegő’s theorem, whose scalar versions are dealt with in the cited paper by the author. The article is accompanied by an exhaustive bibliography of the subject.

MSC:
60G10 Stationary stochastic processes
60G25 Prediction theory (aspects of stochastic processes)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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