A duality method in prediction theory of multivariate stationary sequences

TL;DR

The paper develops a matrix-valued generalization of Nakazi's duality, showing that trigonometric approximation problems in $L^2(W)$ correspond to dual problems in $L^2(W^{-1})$ for multivariate stationary sequences on discrete abelian groups. The core idea is that the map $F o FW$ is an isometric isomorphism between $L^2(W)$ and $L^2(W^{-1})$, enabling explicit duality relations for prediction: the projection and prediction error in $L^2(W)$ can be analyzed via dual objects in $L^2(W^{-1})$. This framework yields concrete prediction formulas, Szegö-type results, and specialized results for $G = Z$, including a multivariate treatment of Nakazi's problem achieved through an approximation strategy when ${ m log}\det W$ is integrable. Overall, the work unifies and extends classical univariate prediction theory to the multivariate setting, with potential applications to multivariate time series analysis and spectral theory of matrix-valued processes.

Abstract

Let W be an integrable positive Hermitian q x q -matrix valued function on the dual group of a discrete abelian group G such that W^{-1} is integrable. Generalizing results of T. Nakazi and of A. G. Miamee and M. Pourahmadi for q=1 we establish a correspondence between trigonometric approximation problems in L^2(W) and certain approximation problems in L^2(W^{-1}). The result is applied to prediction problems for q-variate stationary processes over G, in particular, to the case where G is the group of integers Z.

Theorems & Definitions (22)

• Lemma 3.1
• Theorem 3.2
• Lemma 3.3
• Lemma 3.4
• Remark 3.5
• Theorem 3.6
• Theorem 3.7
• Lemma 3.8
• Theorem 3.9
• Corollary 3.10