Predictive Sets

TL;DR

This paper investigates the properties of predictive sets in stochastic processes, providing conditions for predictivity and exploring connections to harmonic analysis and Gaussian processes.

Contribution

It introduces new criteria for a set to be predictive, extending understanding of predictivity in various classes of stochastic processes.

Findings

Characterization of predictive sets with sufficient and necessary conditions

Analysis of linear predictivity and predictivity among Gaussian processes

Relation of predictive sets to Riesz sets in harmonic analysis

Abstract

A set $P\subset \mathbb N$ is called predictive if for any zero entropy finite-valued stationary process $(X_i)_{i\in \mathbb Z}$, $X_0$ is measurable with respect to $(X_i)_{i\in P}$. We know that $\mathbb N$ is a predictive set. In this paper we give sufficient conditions and necessary ones for a set to be predictive. We also discuss linear predictivity, predictivity among Gaussian processes and relate these to Riesz sets which arise in harmonic analysis.