COMBINING M-DEPENDENCE WITH MARKOVNESS

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)-depe ndent exists if the state space of the sequence has small cardinality. We show that to ensure the existence for the Markov sequences of orde r n = 1 the number of attainable states must be at least m + 2 and tha t this bound is tight. Given a small state space such a sequence exist s 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 f rom our parametric description of all binary m-dependent sequences, m greater than or equal to 0, that are Markov of order 3. (C) Elsevier, Paris.