Compressibility Analysis of Asymptotically Mean Stationary Processes

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 sequences can be approximated by its best k-sparse version under different rates of significant coeffi- cients (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 lp-characterization and we present results for the characterization and analysis of its lp-approximation error function. Furthermore adding ergodicity in the analysis of AMS processes, we have a theorem that shows that its approximation error function is constant and determined in closed-form by the stationary mean of the process. The results and analysis presented in this paper offer a contribution 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.