Continuous Exchangeable Markov Chains, Idempotent and 1-Dependent Copulas

New copula families are constructed based on orthogonality in L2(0,1). Subclasses of idempotent copulas with square integrable densities are derived. It is shown that these copulas generate exchangeable Markov chains that behave as independent and identically distributed random variables conditionally on the initial variable. We prove that the extracted family of copulas is the only set of symmetric idempotent copulas with square integrable densities. We extend these copula families to asymmetric copulas with square integrable densities having special dependence properties. One of our extensions includes the Farlie–Gumbel–Morgenstern (FGM) copula family. The mixing properties of Markov chains generated by these copulas are established. The Spearman’s correlation coefficient ρS is provided for each of these copula families. Some graphs are also provided to illustrate the properties of the copula densities.