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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
##### Keywords:
one-dependence; block factors; stationary process; Markov chain
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