  We characterize absolutely continuous symmetric copulas with square
integrable densities in this paper. This characterization is used to create new
copula families, that are perturbations of the independence copula. The full
study of mixing properties of Markov chains generated by these copula families
is conducted. An extension that includes the Farlie-Gumbel-Morgenstern family
of copulas is proposed. We propose some examples of copulas that generate
non-mixing Markov chains, but whose convex combinations generate $\psi$-mixing
Markov chains. Some general results on $\psi$-mixing are given. The Spearman's
correlation $\rho_S$ and Kendall's $\tau$ are provided for the created copula
families. Some general remarks are provided for $\rho_S$ and $\tau$. A central
limit theorem is provided for parameter estimators in one example. A simulation
study is conducted to support derived asymptotic distributions for some
examples.
