A new result for the analysis of random sequences in terms of ℓp-compressibility is presented. The result characterizes the degree in which an asymptotic mean stationary (AMS) process with the ergodicity property can be approximated by its best k-sparse version under different rates of significant coefficients. Our main theorem shows that the (so called) ℓp-approximation error function of an AMS process has a closed-form expression that is characterized by its stationary mean.