Compressibility Analysis of Asymptotically Mean Stationary Processes

TL;DR

This paper investigates the compressibility of asymptotically mean stationary processes, establishing conditions under which their approximation errors can be characterized and simplified, thus advancing the understanding of sparse process analysis.

Contribution

It introduces the concept of strong $ ext{ell}_p$-characterization for AMS processes and demonstrates how ergodicity simplifies the approximation error analysis.

Findings

AMS processes have a strong $ ext{ell}_p$-characterization.

The approximation error function is constant and explicitly determined by the stationary mean.

Point-wise ergodic theorem is crucial for analyzing compressibility under relaxed assumptions.

Abstract

This work provides new results for the analysis of random sequences in terms of $\ell_p$-compressibility. The results characterize the degree in which a random sequence can be approximated by its best $k$-sparse version under different rates of significant coefficients (compressibility analysis). In particular, the notion of strong $\ell_p$-characterization is introduced to denote a random sequence that has a well-defined asymptotic limit (sample-wise) of its best $k$-term approximation error when a fixed rate of significant coefficients is considered (fixed-rate analysis). The main theorem of this work shows that the rich family of asymptotically mean stationary (AMS) processes has a strong $\ell_p$-characterization. Furthermore, we present results that characterize and analyze the $\ell_p$-approximation error function for this family of processes. Adding ergodicity in the analysis of AMS processes, we introduce a theorem demonstrating that the approximation error function is constant and determined in closed-form by the stationary mean of the process. Our results and analyses contribute to the theory and understanding of discrete-time sparse processes and, on the technical side, confirm how instrumental the point-wise ergodic theorem is to determine the compressibility expression of discrete-time processes even when stationarity and ergodicity assumptions are relaxed.