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：本文描述了具有平方可积密度的绝对连续对称copula。该刻画被用于创建新的copula族，这些族是独立copula的扰动。我们完整研究了这些copula族生成的马尔可夫链的混合性质。提出了一种包含Farlie-Gumbel-Morgenstern copula族的扩展方案。我们给出了一些生成非混合马尔可夫链的copula示例，但其凸组合却能生成ψ-混合马尔可夫链。文中给出了关于ψ-混合的一些一般性结论。对于所创建的copula族，给出了Spearman相关系数ρ_S和Kendall τ的计算公式。提供了关于ρ_S和τ的一些一般性讨论。针对一个示例中的参数估计量，给出了中心极限定理。通过模拟研究支持了某些示例的渐近分布结果。