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Combining $$m$$-dependence with Markovness. (English) Zbl 0910.60055
Summary: Generally, no stationary sequence of random variables which is Markov of order $$n$$ but not of order $$n-1$$ and $$m$$-dependent but not $$(m-1)$$-dependent exists if the state space of the sequence has small cardinality. We show that to ensure the existence for the Markov sequences of order $$n=1$$ the number of attainable states must be at least $$m+2$$ and that this bound is tight. Given a small state space such a sequence exists only for special $$n$$ and $$m$$. On a two-element state space the smallest possible $$n$$ and $$m$$ are shown to be 3 and 2, respectively. This results from our parametric description of all binary $$m$$-dependent sequences, $$m \geq 0$$, that are Markov of order 3.

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60K99 Special processes 60G99 Stochastic processes
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