Abstract
Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.
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References
Aaronson, J., Gilat, D., Keane, M., and Valk, V. de (1989). An algebraic construction of a class of one-dependent processes.Ann. Prob. 17, 128–143.
Chung, K. L. (1967).Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer, Berlin, Heidelberg.
Zygmund, A. (1959).Trigonometric Series, 2nd ed. Cambridge University Press, Cambridge, England.
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Aaronson, J., Gilat, D. & Keane, M. On the structure of 1-dependent Markov chains. J Theor Probab 5, 545–561 (1992). https://doi.org/10.1007/BF01060435
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DOI: https://doi.org/10.1007/BF01060435