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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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