COPULA-BASED CHARACTERIZATIONS FOR HIGHER ORDER MARKOV PROCESSES

In this paper, we obtain characterizations of higher order Markov processes in terms of copulas corresponding to their finite-dimensional distributions. The results are applied to establish necessary and sufficient conditions for Markov processes of a given order to exhibit <italic>m</italic>-dependence, <italic>r</italic>-independence, or conditional symmetry. The paper also presents a study of applicability and limitations of different copula families in constructing higher order Markov processes with the preceding dependence properties. We further introduce new classes of copulas that allow one to combine Markovness with <italic>m</italic>-dependence or <italic>r</italic>-independence in time series.