Ergodicity of stationary Markov renewal process with finite states

This is a sequel of the author’s previous paper. There, the mathematical definition of the stationary Markov renewal process (MRP) was given with a complete proof that the defined process is really a stationary point process on the whole time axis. As a consequence, various stochastic processes appearing in practice which are derived from MRP can easily be shown to be stationary. The author has been interested in the calculation to the power spectrum of MRP generated signals in electrical or nervous systems. For this purpose, it was necessary to construct a stationary process generated by MRP and hence it was inevitable to introduce a stationary MRP on the whole time axis. In this paper, the ergodicity of the one-parameter group of measure-preserving transformations associated with the time shift of the stationary MRP is discussed. An irreducible MRP is either aperiodic or periodic with some period. Here, a proof is given that an aperiodic MRP is strongly mixing, from which ergodicity follows. On the contrary, as a periodic MRP is only ergodic without being mixing, it is directly proved ergodic. Once the ergodicity of MRP is established, various stationary processes derived thereof are easily shown to be ergodic as well. This means that the characteristics, among which the power spectrum of each sample process is our concern, are sample-wise invariant, which will be useful in many applications. Recently, as MRP and related stochastic processes are widely used in many fields, e.g. physics, OR, communication theory, and bioscience, our results will find large utilities in both the theory and the application of stochastic processes.