The role of the time-arrow in mean-square estimation of stochastic processes

TL;DR

This paper explores how the past and future data of multivariate stochastic processes differently influence the present, revealing a time-directional determinism linked to the geometry of the shift-operator.

Contribution

It characterizes all rank-one regular processes that are deterministic in reverse time, connecting classical theories with modern control engineering insights.

Findings

Determinism in reverse time depends on process rank and geometry.

Scalar processes do not exhibit this time-directional dichotomy.

Provides a geometric framework for understanding stochastic process time asymmetry.

Abstract

The purpose of this paper is to explain a certain dichotomy between the information that the past and future values of a multivariate stochastic process carry about the present. More specifically, vector-valued, second-order stochastic processes may be deterministic in one time-direction and not the other. This phenomenon, which is absent in scalar-valued processes, is deeply rooted in the geometry of the shift-operator. The exposition and the examples we discuss are based on the work of Douglas, Shapiro and Shields on cyclic vectors of the backward shift and relate to classical ideas going back to Wiener and Kolmogorov. We focus on rank-one stochastic processes for which we present a characterization of all regular processes that are deterministic in the reverse time-direction. The paper builds on examples and the goal is to provide pertinent insights to a control engineering audience.