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On the structure of 1-dependent Markov chains. (English) Zbl 0754.60070
A stochastic process $${\mathcal W}=\{W_ n\}^ \infty_{n=1}$$ is a 2-block factor if there are independent, identically distributed random variables $$\{X_ n\}^ \infty_{n=1}$$ and a measurable function $$f$$ of two real variables such that $$W_ n=f(X_ n,X_{n+1})$$ for all $$n\geq 1$$. Clearly, a 2-block factor is stationary and 1-dependent in the sense that $$(W_ 1,W_ 2,\dots,W_ n)$$ and $$(W_{n+2},W_{n+3},\dots)$$ are independent for all $$n\geq 1$$. Assume now that $$\mathcal W$$ is stationary and 1-dependent. The authors give two examples, including a Markov chain with five states, to show that $$\mathcal W$$ need not be a 2-block factor. They also show that if $$\mathcal W$$ is a Markov chain with four or fewer states, or a renewal process, then $$\mathcal W$$ will be a 2-block factor.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces
##### Keywords:
2-block factor; renewal process
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##### References:
 [1] 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. · Zbl 0681.60038 [2] Chung, K. L. (1967).Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer, Berlin, Heidelberg. · Zbl 0146.38401 [3] Zygmund, A. (1959).Trigonometric Series, 2nd ed. Cambridge University Press, Cambridge, England. · Zbl 0085.05601
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