New copula families and mixing properties

TL;DR

This paper introduces new symmetric copula families with square integrable densities, studies their mixing properties, and explores their applications in Markov chains, including extensions and examples demonstrating various mixing behaviors.

Contribution

It characterizes symmetric copulas with square integrable densities, creates new copula families as perturbations of independence, and analyzes their mixing properties and statistical measures.

Findings

New copula families with explicit Spearman's and Kendall's measures

Examples of non-mixing and $ ext{psi}$-mixing Markov chains

Central limit theorem for parameter estimators

Abstract

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.