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An Invariance Principle for Fractional Brownian Sheets

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  • Yizao Wang

    (University of Cincinnati)

Abstract

We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (Bernoulli 17:88–113, 2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an $$m$$ -approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (Stoch. Process. Appl. 123:1–14, 2013).

Suggested Citation

  • Yizao Wang, 2014. "An Invariance Principle for Fractional Brownian Sheets," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1124-1139, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0483-2
    DOI: 10.1007/s10959-013-0483-2
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    References listed on IDEAS

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    1. Gabriel Lang & Philippe Soulier, 2000. "Convergence de mesures spectrales aléatoires et applications à des principes d'invariance," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 41-51, January.
    2. Jérôme Dedecker & Florence Merlevède & Dalibor Volný, 2007. "On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 20(4), pages 971-1004, December.
    3. El Machkouri, Mohamed & Volný, Dalibor & Wu, Wei Biao, 2013. "A central limit theorem for stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 1-14.
    4. Florence Merlevède & Magda Peligrad, 2006. "On the Weak Invariance Principle for Stationary Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 19(3), pages 647-689, December.
    5. Frédéric Lavancier, 2007. "Invariance principles for non-isotropic long memory random fields," Statistical Inference for Stochastic Processes, Springer, vol. 10(3), pages 255-282, October.
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    Cited by:

    1. Ulrich K. Müller & Mark W. Watson, 2022. "Spatial Correlation Robust Inference," Econometrica, Econometric Society, vol. 90(6), pages 2901-2935, November.

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