[Paper Review] Multivariate prediction and matrix Szegö theory

Ν. H. Bingham|arXiv (Cornell University)|Mar 5, 2012

Mathematical functions and polynomials46 references4 citations

TL;DR

This paper extends classical univariate prediction theory and Szegö's theorem to the multivariate setting using matrix orthogonal polynomials on the unit circle (MOPUC). It establishes key results such as Verblunsky's theorem for matrix measures, matrix Szegö's theorem linking spectral density to Verblunsky coefficients, and characterizations of regularity conditions via matrix Muckenhoupt and Helson-Szegö conditions.

ABSTRACT

Following the recent survey by the same author of Szegö's theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).

Motivation & Objective

• To extend univariate prediction theory and Szegö's theorem to multivariate stationary processes using matrix orthogonal polynomials on the unit circle (MOPUC).
• To establish a matrix version of Verblunsky's theorem, showing a bijection between matrix Verblunsky coefficients and matrix spectral measures.
• To characterize regularity conditions—such as pure minimality, positive angle, and complete regularity—using matrix spectral density and Muckenhoupt-type conditions.
• To connect multivariate prediction theory with Hankel operators and the matrix Nehari problem, extending classical operator-theoretic tools to the matrix setting.

Proposed method

• Uses the Kolmogorov isomorphism theorem to represent stationary vector-valued processes via spectral measures on the unit circle.
• Applies Bernstein-Szegö approximation to derive matrix versions of Szegö's theorem, linking the infinite product of determinants of $1 - a_n^\dagger a_n$ to the integral of $\operatorname{tr} \log w(\theta)$.
• Employs matrix spectral factorization and matrix Szegö functions to analyze the spectral density $W(\theta)$ and its regularity.
• Characterizes the positive angle condition via the matrix Muckenhoupt condition $(A_2)$: $\sup_I \left\| \left( \frac{1}{|I|} \int_I W \right)^{1/2} \left( \frac{1}{|I|} \int_I W^{-1} \right)^{1/2} \right\| < \infty$.
• Extends the concept of complete regularity to the matrix case via the condition $\limsup_{|I| \to 0} \left\| \left( \frac{1}{|I|} \int_I W \right)^{1/2} \left( \frac{1}{|I|} \int_I W^{-1} \right)^{1/2} \right\| < \infty$.
• Utilizes Hankel operators and their connections to the matrix Nehari problem to study prediction error and regularity in multivariate time series.

Experimental results

Research questions

• RQ1 What is the matrix analog of Verblunsky's theorem, and how does it parametrize matrix spectral measures?
• RQ2 How can Szegö's theorem be extended to matrix-valued spectral measures, and what is the role of the Verblunsky coefficients in this extension?
• RQ3 What matrix conditions correspond to the scalar regularity conditions such as pure minimality, positive angle, and complete regularity?
• RQ4 How do the matrix Muckenhoupt condition and the Helson-Szegö condition relate to the regularity of multivariate prediction processes?
• RQ5 What is the matrix version of Baxter's theorem, and how can it be derived from matrix Hankel operators and the Nehari problem?

Key findings

• Verblunsky's theorem holds for MOPUC: any sequence of $\ell \times \ell$ matrices $a_n$ with $\|a_n\| < 1$ corresponds uniquely to a matrix spectral measure $\mu$.
• Matrix Szegö's theorem states that $\sum_{n=1}^\infty \|a_n^\dagger a_n\| < \infty$ if and only if the integral $\int \operatorname{tr} \log w(\theta) \, d\theta / 2\pi > -\infty$, defining the class of Szegö measures.
• The positive angle condition $(PA)$ is equivalent to the matrix Muckenhoupt condition $(A_2)$, which requires $\sup_I \left\| \left( \frac{1}{|I|} \int_I W \right)^{1/2} \left( \frac{1}{|I|} \int_I W^{-1} \right)^{1/2} \right\| < \infty$.
• Complete regularity is equivalent to the strengthened $\limsup$ version of the Muckenhoupt condition: $\limsup_{|I| \to 0} \left\| \left( \frac{1}{|I|} \int_I W \right)^{1/2} \left( \frac{1}{|I|} \int_I W^{-1} \right)^{1/2} \right\| < \infty$.
• The matrix version of the Helson-Lowdenslager theorem holds: $\exp \left( \frac{1}{\ell} \int \operatorname{tr} \log w(\theta) \, d\theta / 2\pi \right) = \inf_{A,P} \int \frac{1}{\ell} \operatorname{tr} \left[ (A+P)^\dagger d\mu (A+P) \right]$.
• Hankel operators are linked to the matrix Nehari problem and the Muckenhoupt condition, providing a bridge between prediction theory and operator theory in the multivariate case.