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Convergence rates in the functional CLT for α-mixing triangular arrays

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  • Hafouta, Yeor

Abstract

We obtain convergence rates (in the Levi–Prokhorove metric) in the functional central limit theorem (CLT) for partial sums Sn=∑j=1nξj,n of triangular arrays {ξ1,n,ξ2,n,…,ξn,n} satisfying some mixing and moment conditions (which are not necessarily uniform in n). For certain classes of additive functionals of triangular arrays of contracting Markov chains (in the sense of Dobrushin) we obtain rates which are close to the best rates obtained for independent random variables. In addition, we obtain close to optimal rates in the usual CLT and a moderate deviations principle and some Rosenthal type inequalities. We will also discuss applications to some classes of local statistics (e.g. covariance estimators), as well as expanding non-stationary dynamical systems, which can be reduced to non-uniformly mixing triangular arrays by an approximation argument. The main novelty here is that our results are obtained without any assumptions about the growth rate of the variance of Sn. The result are obtained using a certain type of block decomposition, which, in a sense, reduces the problem to the case when the variance of Sn is “not negligible” in comparison with the (new) number of summands.

Suggested Citation

  • Hafouta, Yeor, 2023. "Convergence rates in the functional CLT for α-mixing triangular arrays," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 242-290.
  • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:242-290
    DOI: 10.1016/j.spa.2023.04.008
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    References listed on IDEAS

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    1. Merlevède, Florence & Peligrad, Magda, 2020. "Functional CLT for nonstationary strongly mixing processes," Statistics & Probability Letters, Elsevier, vol. 156(C).
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
    3. Francq, Christian & Zakoïan, Jean-Michel, 2005. "A Central Limit Theorem For Mixing Triangular Arrays Of Variables Whose Dependence Is Allowed To Grow With The Sample Size," Econometric Theory, Cambridge University Press, vol. 21(6), pages 1165-1171, December.
    4. Doukhan, Paul & Neumann, Michael H., 2007. "Probability and moment inequalities for sums of weakly dependent random variables, with applications," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 878-903, July.
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