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Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications

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  • Chen, Xiaohong
  • White, Halbert

Abstract

To obtain consistency results for nonparametric estimators based on stochastic processes relevant in econometrics, we introduce the notions of Hilbert space-valued Lp mixingales and near-epoch dependent arrays, and we prove weak and strong laws of large numbers by using a new exponential inequality for Hilbert (H) space-valued martingale difference arrays. We follow Andrews (1988, Econometric Theory 4, 458–467), Hansen (1991, Econometric Theory 7, 213–221; 1992, Econometric Theory 8, 421–422), Davidson (1993, Statistics and Probability Letters 16,301–304), and de Jong (1995, Econometric Theory 11, 347–358), extending results for H = R and improving memory conditions in certain instances. We give as examples consistency results for series and kernel estimators.

Suggested Citation

  • Chen, Xiaohong & White, Halbert, 1996. "Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications," Econometric Theory, Cambridge University Press, vol. 12(2), pages 284-304, June.
    • Handle: RePEc:cup:etheor:v:12:y:1996:i:02:p:284-304_00
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    Cited by:

    1. Escanciano, J. Carlos & Velasco, Carlos, 2006. "Generalized spectral tests for the martingale difference hypothesis," Journal of Econometrics, Elsevier, vol. 134(1), pages 151-185, September.
    2. Chen Xiaohong & White Halbert, 2002. "Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 6(1), pages 1-55, April.
    3. Chen, Xiaohong & White, Halbert, 1998. "Nonparametric Adaptive Learning with Feedback," Journal of Economic Theory, Elsevier, vol. 82(1), pages 190-222, September.
    4. Gábor Lugosi & Shie Mannor & Gilles Stoltz, 2008. "Strategies for Prediction Under Imperfect Monitoring," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 513-528, August.
    5. Shao, Xiaofeng, 2011. "A bootstrap-assisted spectral test of white noise under unknown dependence," Journal of Econometrics, Elsevier, vol. 162(2), pages 213-224, June.
    6. Escanciano, J. Carlos & Velasco, Carlos, 2006. "Testing the martingale difference hypothesis using integrated regression functions," Computational Statistics & Data Analysis, Elsevier, vol. 51(4), pages 2278-2294, December.
    7. Hörner, Johannes & Takahashi, Satoru, 2016. "How fast do equilibrium payoff sets converge in repeated games?," Journal of Economic Theory, Elsevier, vol. 165(C), pages 332-359.
    8. Vladimir V'yugin, 2014. "Log-Optimal Portfolio Selection Using the Blackwell Approachability Theorem," Papers 1410.5996, arXiv.org, revised Jun 2015.
    9. Shie Mannor & Gilles Stoltz, 2009. "A Geometric Proof of Calibration," Working Papers hal-00442042, HAL.
    10. Hsu, Shih-Hsun & Kuan, Chung-Ming, 2011. "Estimation of conditional moment restrictions without assuming parameter identifiability in the implied unconditional moments," Journal of Econometrics, Elsevier, vol. 165(1), pages 87-99.
    11. Juan Carlos Escanciano, 2010. "The Integrated Instrumental Variables Estimator: Exploiting Nonlinearities for Identification of Linear Models," CAEPR Working Papers 2010-001, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    12. Paulo M.D.C. Parente & Richard J. Smith, 2018. "Generalised Empirical Likelihood Kernel Block Bootstrapping," Working Papers REM 2018/55, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    13. Escanciano, Juan Carlos & Jacho-Chávez, David T., 2010. "Approximating the critical values of Cramér-von Mises tests in general parametric conditional specifications," Computational Statistics & Data Analysis, Elsevier, vol. 54(3), pages 625-636, March.
    14. Shie Mannor & Gilles Stoltz, 2010. "A Geometric Proof of Calibration," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 721-727, November.

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