Markov Random Processes Are Neither Bandlimited nor Recoverable From Samples or After Quantization

This paper considers basic questions regarding Markov random processes. It shows that continuous-time, continuous-valued, wide-sense stationary, Markov processes that have absolutely continuous second-order distribution and finite second moment are not bandlimited. It also shows that continuous-time, stationary, Markov processes that are continuous-valued or discrete-valued and satisfy additional mild conditions cannot be recovered from uniform sampling. Further it shows that continuous-time, continuous-valued, stationary, Markov processes that have absolutely continuous second-order distributions and are continuous almost surely, cannot be recovered without error after quantization. Finally, it provides necessary and sufficient conditions for stationary, discrete-time, Markov processes to have zero entropy rate, and relates this to information singularity.