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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
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[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
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[3] Zygmund, A. (1959).Trigonometric Series, 2nd ed. Cambridge University Press, Cambridge, England. · Zbl 0085.05601
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