Markov Random Processes are not Recoverable After Quantization and Mostly not Recoverable From Samples

Abstract

Markov random processes and general random processes are considered. It is shown that continuous-time, continuous-valued, wide-sense stationary, Markov random processes that have absolutely continuous second order distributions are not bandlimited. It is also shown that when these processes are strictly stationary and continuous almost surely, they cannot be recovered without error from their quantized versions. Further, it is shown that continuous-time, discrete-valued Markov random processes, which are uniformly bounded and satisfy an additional condition, can be recovered with zero average distortion from an appropriate set of samples for a general distortion measure. A similar result is shown for general continuous-time random processes with r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> power distortion measure. Additionally, it is shown that under a milder condition on the Markov processes and a different condition on the sampling times (e.g., uniform sampling), such processes cannot be recovered with zero average distortion. Finally, the notion of information-singularity is extended to continuous-time random processes, and it is shown that both continuous- and discrete-time Markov processes are not information-singular.