Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions

Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions

2005 | Richard C. Bradley
This paper provides an updated and expanded survey of basic properties of strong mixing conditions, including new developments and open questions. It introduces various measures of dependence, such as α, φ, ψ, ρ, β, ψ*, ψ', and I, which quantify the dependence between σ-fields. These measures have specific ranges and relationships, with some being equivalent to independence. The paper also discusses different types of strong mixing conditions, such as α-mixing, φ-mixing, ψ-mixing, ρ-mixing, and β-mixing, and their relationships. It highlights the asymmetry of φ-mixing and introduces interlaced strong mixing conditions. The paper also explores conditional mixing and two-part mixing conditions, as well as tail σ-fields and their implications for ergodicity. It discusses the behavior of dependence coefficients in strictly stationary sequences and their limits. The paper also addresses open questions, such as whether certain mixing conditions imply others, and whether strictly stationary sequences can be represented as instantaneous functions of Harris recurrent Markov chains. The paper concludes with a discussion of geometric ergodicity and Doeblin's condition, and their implications for mixing and convergence.This paper provides an updated and expanded survey of basic properties of strong mixing conditions, including new developments and open questions. It introduces various measures of dependence, such as α, φ, ψ, ρ, β, ψ*, ψ', and I, which quantify the dependence between σ-fields. These measures have specific ranges and relationships, with some being equivalent to independence. The paper also discusses different types of strong mixing conditions, such as α-mixing, φ-mixing, ψ-mixing, ρ-mixing, and β-mixing, and their relationships. It highlights the asymmetry of φ-mixing and introduces interlaced strong mixing conditions. The paper also explores conditional mixing and two-part mixing conditions, as well as tail σ-fields and their implications for ergodicity. It discusses the behavior of dependence coefficients in strictly stationary sequences and their limits. The paper also addresses open questions, such as whether certain mixing conditions imply others, and whether strictly stationary sequences can be represented as instantaneous functions of Harris recurrent Markov chains. The paper concludes with a discussion of geometric ergodicity and Doeblin's condition, and their implications for mixing and convergence.
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