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.

翻译：本文刻画了具有平方可积密度的绝对连续对称copula。这一刻画被用于构造新的copula族，这些族是独立copula的扰动。我们全面研究了由这些copula族生成的马尔可夫链的混合性质。提出了一个包含Farlie-Gumbel-Morgenstern copula族的推广。我们给出了一些生成非混合马尔可夫链的copula例子，但它们的凸组合却能生成ψ-混合马尔可夫链。给出了关于ψ-混合的一些一般性结果。为所构造的copula族提供了Spearman相关系数ρ_S和Kendall's τ。关于ρ_S和τ给出了一些一般性注释。在一个例子中，为参数估计量提供了中心极限定理。通过模拟研究来支持某些例子的渐近分布。