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On 1-dependent processes and \(k\)-block factors. (English) Zbl 0788.60049
Summary: A stationary process \((X_ n)_{n\in \mathbb{Z}}\) is said to be \(k\)- dependent if \(\{X_ n\}_{n < 0}\) is independent of \(\{X_ n\}_{n > k - 1}\). It is said to be a \(k\)-block factor on a process \(\{Y_ n\}\) if it can be represented as \(X_ n = f(Y_ n,\dots\), \(Y_{n + k - 1})\), where \(f\) is a measurable function of \(k\) variables. Any \((k + 1)\)-block factor of an i.i.d. process is \(k\)-dependent. We answer an old question by showing that there exists a one-dependent process which is not a \(k\)- block factor of any i.i.d. process for any \(k\). Our method also leads to generalizations of this result and to a simple construction of an eight- state one-dependent Markov chain which is not a two-block factor of an i.i.d. process.

MSC:
60G10 Stationary stochastic processes
54H20 Topological dynamics (MSC2010)
28D05 Measure-preserving transformations
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