On predictors for band-limited and high-frequency time series

TL;DR

This paper investigates the predictability of certain classes of discrete time processes, demonstrating that band-limited and high-frequency processes are predictable using convolution sum approximations, with implications for signal processing.

Contribution

It introduces a novel approach to predictability based on convolution sums over past data and characterizes classes of processes that are predictable in this framework.

Findings

All band-limited processes are predictable.

High-frequency processes with zero low-frequency energy are predictable.

Predictability extends to mixed processes with an ideal low-pass filter.

Abstract

Pathwise predictability and predictors for discrete time processes are studied in deterministic setting. It is suggested to approximate convolution sums over future times by convolution sums over past time. It is shown that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. In addition, a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.