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Approximately mixing time series

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  • Kutta, Tim

Abstract

In this note, we present the new concept of approximate mixing for random variables on metric spaces. Approximate mixing is characterized by two constants ϵ,δ≥0, where ϵ is the mixing coefficient and δ is a slack variable. In the case δ=0, approximate mixing reduces to classical β-mixing. For positive slack, δ>0, it becomes more general than traditional mixing assumptions, including important time series such as autoregressive processes on Hilbert spaces, that are generally not mixing. We prove that under approximate mixing analogous covariance inequalities hold as in the mixing case. We use these results to prove a central limit theorem for non-stationary time series on Hilbert spaces, which has potential applications in functional data analysis.

Suggested Citation

  • Kutta, Tim, 2025. "Approximately mixing time series," Statistics & Probability Letters, Elsevier, vol. 220(C).
    • Handle: RePEc:eee:stapro:v:220:y:2025:i:c:s0167715225000069
    DOI: 10.1016/j.spl.2025.110360
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    References listed on IDEAS

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    1. Mokkadem, Abdelkader, 1988. "Mixing properties of ARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 29(2), pages 309-315, September.
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    3. Nze, Patrick Ango & Doukhan, Paul, 2004. "Weak Dependence: Models And Applications To Econometrics," Econometric Theory, Cambridge University Press, vol. 20(6), pages 995-1045, December.
    4. Coulon-Prieur, Clémentine & Doukhan, Paul, 2000. "A triangular central limit theorem under a new weak dependence condition," Statistics & Probability Letters, Elsevier, vol. 47(1), pages 61-68, March.
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