A finite-state Markov chain whose state mean is nonzero has an autocorrelation that does not decay to zero, hence is not Lp integrable; the paper's derivation for the two-state case is incorrect.
D¸ ebowski, On processes with hyperbolically decaying autocorrelations, Journal of Time Series Analysis 32 (20 11) 580–584
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Autocorrelation Function Characterization of Continuous Time Markov Chains
A finite-state Markov chain whose state mean is nonzero has an autocorrelation that does not decay to zero, hence is not Lp integrable; the paper's derivation for the two-state case is incorrect.