Precise best k-term approximation error analysis of ergodic processes

The characterization of ℓ_p-compressible random sequences is revisited and extended to the case of stationary and ergodic processes. The main result of this work offers a simple-to-check necessary and sufficient condition for a stationary and ergodic sequence to be ℓ_p-compressible in the sense proposed by Amini, Unser and Marvasti [1, Def. 6]. Furthermore, for non ℓ_p-compressible random sequences, we provide a closed-form expression for the best k-term relative approximation error given a rate of coefficients as the block-length tends to infinity.