  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.
