Applications of a finite-dimensional duality principle to some prediction problems

TL;DR

This paper introduces a finite-dimensional duality lemma that simplifies and unifies key results in prediction theory and time series analysis, especially for cases with few missing values, without requiring stationarity.

Contribution

The paper presents a new finite-dimensional duality lemma that unifies various prediction results and extends applicability to nonstationary processes using elementary linear algebra.

Findings

Reveals the finite-dimensional nature of certain prediction problems

Provides a unified approach using duality and biorthogonality

Applicable to nonstationary processes without stationarity assumptions

Abstract

Some of the most important results in prediction theory and time series analysis when finitely many values are removed from or added to its infinite past have been obtained using difficult and diverse techniques ranging from duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear regression in statistics (Box and Tiao, 1975). We unify these results via a finite-dimensional duality lemma and elementary ideas from the linear algebra. The approach reveals the inherent finite-dimensional character of many difficult prediction problems, the role of duality and biorthogonality for a finite set of random variables. The lemma is particularly useful when the number of missing values is small, like one or two, as in the case of Kolmogorov and Nakazi prediction problems. The stationarity of the underlying process is not a requirement. It opens up the possibility of extending such results to nonstationary processes.