Effects of downsampling on statistics of discrete-time semi-Markov processes

In this paper, we first present a novel approach to finding the statistics of the downsampled sequence of a discrete time semi-Markov process in terms of the sojourn time distributions and states transition matrix of the resulting process. Moreover, we further show that the statistics of the original semi-Markov process cannot be uniquely determined given the downsampled sequence. This suggests a singularity issue resulting from the downsampling regardless of the bandwidth of the original process. Numerical results based on derived theoretical investigation have been further verified using simulations.