On mixtures of copulas and mixing coefficients

Abstract

Please cite this article as: M. Longla, On mixtures of copulas and mixing coefficients, Journal of Multivariate Analysis (2015), http://dx. AbstractWe show that if the density of the absolutely continuous part of a copula is bounded away from zero on a set of Lebesgue measure 1, then that copula generates "lower ψ-mixing" stationary Markov chains. This conclusion implies φ-mixing, ρ-mixing, β-mixing and "interlaced ρ-mixing". We also provide some new results on the mixing structure of Markov chains generated by mixtures of copulas.

On dependence structure of copula-based Markov chains

“…This property has been used by many authors to establish mixing properties of copulabased Markov chains. We can cite Longla (2015), Longla (2014), Longla and Peligrad (2012) who provided some results for reversible Markov chains and Beare (2010) who presented results for ρ-mixing. We are assuming in the sequel that variables in a copulabased Markov chain have the uniform distribution on [0, 1].…”

“…A non-strict Archimedean copula is defined by some convex function ϕ where A = σ(X i , i ≤ 0), B = σ(X i , i ≥ n) and P is the defined probability measure. For Markov chains generated by an absolutely continuous copula (see Longla (2013) or Longla (2015)) these coefficients are…”

“…He also includes a detailed list of relationships between different mixing coefficients. Longla (2015) showed that Markov chains generated by copulas are lower ψ-mixing when the density of the absolutely continuous part of the generating copula is bounded away from zero on a set of Lebesgue measure 1. Longla and Peligrad (2012) defined the mixing coefficients for an absolutely continuous copula and an absolutely continuous invariant distribution for the states of a stationary Markov chain that they generate.…”

On some mixing properties of copula-based Markov chains

“…As a result of Theorem 2.1.3, following Longla (2015), based on the fact that the density of the copula C n θ,Π (u, v) is bounded away from zero on a set of Lebesgue Measure 1, we can conclude the following:…”

“…This property has been used by many authors to establish mixing properties of copula-based Markov chains. We can cite Longla (2015), Longla (2014), Longla and Peligrad (2012) who provided some results for reversible Markov chains, Beare (2010) who presented results for ρ-mixing among others.…”