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Central limit theorem under the Dedecker–Rio condition in some Banach spaces

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  • Bigot, Aurélie

Abstract

We extend the central limit theorem under the Dedecker–Rio condition to adapted stationary and ergodic sequences of random variables taking values in a class of smooth Banach spaces. This result applies to the case of random variables taking values in Lp(μ), with 2⩽p<∞ and μ a σ-finite real measure. As an application we give a sufficient condition for empirical processes indexed by Sobolev balls to satisfy the central limit theorem, and discuss about the optimality of these conditions.

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  • Bigot, Aurélie, 2024. "Central limit theorem under the Dedecker–Rio condition in some Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:spapps:v:176:y:2024:i:c:s030441492400125x
    DOI: 10.1016/j.spa.2024.104419
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    References listed on IDEAS

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    1. Dedecker, J. & Merlevède, F., 2015. "Moment bounds for dependent sequences in smooth Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3401-3429.
    2. Dedecker, Jérôme & Merlevède, Florence, 2003. "The conditional central limit theorem in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 229-262, December.
    3. Dedecker, Jérôme & Doukhan, Paul, 2003. "A new covariance inequality and applications," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 63-80, July.
    4. Richard C. Bradley, 1997. "On Quantiles and the Central Limit Question for Strongly Mixing Sequences," Journal of Theoretical Probability, Springer, vol. 10(2), pages 507-555, April.
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