Copulas for Markovian Dependence

Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper shows some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete coand countermonotonicity and independence (Fréchet copulas) are shown to imply quite a restricted type of Markov process, and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable, or equivalently, conditionally i.i.d.