Remarks on the speed of convergence of mixing coefficients and applications

Abstract

In this paper, we study dependence coefficients for copula-based Markov chains. We provide new tools to check the convergence rates of mixing coefficients of copula-based Markov chains. We study Markov chains generated by the Metropolis-hastings algorithm and give conditions on the proposal that ensure exponential ρ-mixing, β-mixing and φ-mixing. A general necessary condition on symmetric copulas to generate exponential ρ-mixing or φ-mixing is given. At the end of the paper, we comment and improve some of our previous results on mixtures of copulas.

On mixtures of copulas and mixing coefficients

“…We have initially shown in Longla and Peligrad (2012) a result on mixtures of copulas for absolute regularity. A second result on ρ-mixing for mixtures of copulas was provided in Longla (2013). Here, we extend the results to other mixing coefficients that are not less important.…”

On dependence structure of copula-based Markov chains

“…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…”

“…This work is motivated by applications in Bayesian analysis of Monte Carlo Markov chains. Longla and Peligrad (2012), Longla (2013) have provided several theorems on exponential ρ-mixing and geometric ergodicity of convex combinations of geometrically ergodic Markov chains. This work completes the ideas provided in the two cited papers, that one can read for more information on copulas and their importance in assessing the dependence structure of Markov chains.…”

On some basic features of strictly stationary, reversible Markov chains

“…It is a well known feature of Theorem 1.1 that the geometric ergodicity and spectral gap conditions therein can be formulated in terms of the dependence coefficients associated with the absolute regularity (‐mixing) and ‐mixing conditions respectively. See for example the articles of Longla and Peligrad (2012), Longla (2013) and Longla (2014). It turns out that the dependence coefficients associated with the strong mixing (‐mixing) condition of Rosenblatt (1956) apparently can bring some extra clarity to, and thereby perhaps facilitate a relatively gentle exposition of, Theorem 1.1.…”