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A central limit theorem for stationary random fields

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  • El Machkouri, Mohamed
  • Volný, Dalibor
  • Wu, Wei Biao

Abstract

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form Xk=g(εk−s,s∈Zd), k∈Zd, where (εi)i∈Zd are iid random variables and g is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:1:p:1-14
    DOI: 10.1016/j.spa.2012.08.014
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    References listed on IDEAS

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    1. Paulauskas, Vygantas, 2010. "On Beveridge-Nelson decomposition and limit theorems for linear random fields," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 621-639, March.
    2. El Machkouri, Mohamed, 2002. "Kahane-Khintchine inequalities and functional central limit theorem for stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 285-299, December.
    3. Jenish, Nazgul & Prucha, Ingmar R., 2009. "Central limit theorems and uniform laws of large numbers for arrays of random fields," Journal of Econometrics, Elsevier, vol. 150(1), pages 86-98, May.
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    2. Hagemann, Andreas, 2019. "Placebo inference on treatment effects when the number of clusters is small," Journal of Econometrics, Elsevier, vol. 213(1), pages 190-209.
    3. Koch, Erwan & Dombry, Clément & Robert, Christian Y., 2019. "A central limit theorem for functions of stationary max-stable random fields on Rd," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3406-3430.
    4. Bai, Jushan & Han, Xu & Shi, Yutang, 2020. "Estimation and inference of change points in high-dimensional factor models," Journal of Econometrics, Elsevier, vol. 219(1), pages 66-100.
    5. Peligrad, Magda & Zhang, Na, 2018. "On the normal approximation for random fields via martingale methods," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1333-1346.
    6. Timothy Fortune & Magda Peligrad & Hailin Sang, 2021. "A local limit theorem for linear random fields," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 696-710, September.
    7. Andreas Hagemann, 2020. "Inference with a single treated cluster," Papers 2010.04076, arXiv.org.
    8. Bucchia, Béatrice & Wendler, Martin, 2017. "Change-point detection and bootstrap for Hilbert space valued random fields," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 344-368.
    9. Chen, Likai & Wang, Weining & Wu, Wei Biao, 2019. "Inference of Break-Points in High-Dimensional Time Series," IRTG 1792 Discussion Papers 2019-013, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    10. Lin, Han-Mai & Merlevède, Florence, 2022. "On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 198-220.
    11. Klicnarová, Jana & Volný, Dalibor & Wang, Yizao, 2016. "Limit theorems for weighted Bernoulli random fields under Hannan’s condition," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1819-1838.
    12. Likai Chen & Weining Wang & Wei Biao Wu, 2017. "Dynamic Semiparametric Factor Model with a Common Break," SFB 649 Discussion Papers SFB649DP2017-026, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    13. Zhang, Rongmao & Chan, Ngai Hang & Chi, Changxiong, 2023. "Nonparametric testing for the specification of spatial trend functions," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    14. Kurisu, Daisuke, 2019. "On nonparametric inference for spatial regression models under domain expanding and infill asymptotics," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.
    15. Wang, Yizao & Woodroofe, Michael, 2014. "On the asymptotic normality of kernel density estimators for causal linear random fields," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 201-213.
    16. Lahiri, S.N. & Robinson, Peter M., 2016. "Central limit theorems for long range dependent spatial linear processes," LSE Research Online Documents on Economics 65331, London School of Economics and Political Science, LSE Library.
    17. Tempelman, Arkady, 2022. "Randomized multivariate Central Limit Theorems for ergodic homogeneous random fields," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 89-105.
    18. Andreas Hagemann, 2019. "Permutation inference with a finite number of heterogeneous clusters," Papers 1907.01049, arXiv.org, revised Feb 2023.
    19. Volný, Dalibor & Wang, Yizao, 2014. "An invariance principle for stationary random fields under Hannan’s condition," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4012-4029.
    20. Bouzebda, Salim & Slaoui, Yousri, 2019. "Large and moderate deviation principles for recursive kernel estimators of a regression function for spatial data defined by stochastic approximation method," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 17-28.

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