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.

这项工作为随机序列的分析提供了新的结果 ${\ell }_{磷}$- 可压缩性。结果表征了随机序列在不同的显着系数率（可压缩性分析）下可以通过其最佳k稀疏版本近似的程度。特别是强的概念${\ell }_{磷}$- 特征被引入来表示一个随机序列，当考虑一个固定比率的显着系数（固定比率分析）时，它的最佳k项近似误差具有明确定义的渐近极限（样本方式）。这项工作的主要定理表明，渐近平均平稳 (AMS) 过程的富族具有很强的${\ell }_{磷}$- 表征。此外，我们提出了表征和分析${\ell }_{磷}$- 这一系列过程的近似误差函数。在 AMS 过程的分析中添加遍历性，我们引入了一个定理，证明近似误差函数是恒定的，并且由过程的平稳平均值以封闭形式确定。我们的结果和分析有助于对离散时间稀疏过程的理论和理解，并且在技术方面，确认了逐点遍历定理对于确定离散时间过程的可压缩性表达式有多么重要，即使在平稳性和遍历性假设是轻松。