A duality method in prediction theory of multivariate stationary sequences

TL;DR

This paper develops a duality approach in the prediction theory of multivariate stationary sequences, linking approximation problems in weighted L^2 spaces and applying it to multivariate prediction over discrete groups.

Contribution

It generalizes existing univariate results to multivariate cases and establishes a duality framework for prediction problems in multivariate stationary processes.

Findings

Established a duality between approximation problems in L^2(W) and L^2(W^{-1})

Extended univariate prediction results to multivariate stationary sequences

Applied the duality to prediction problems over the integers Z

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.