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The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

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  • Li, Deli
  • Bhaskara Rao, M.
  • Tomkins, R. J.

Abstract

The Chung-Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented.

Suggested Citation

  • Li, Deli & Bhaskara Rao, M. & Tomkins, R. J., 2001. "The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 191-217, August.
    • Handle: RePEc:eee:jmvana:v:78:y:2001:i:2:p:191-217
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    References listed on IDEAS

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    1. Li, D. L. & Rao, M. B. & Wang, X. C., 1995. "On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 181-198, February.
    2. Li, Deli & Tomkins, R. J., 1996. "Laws of the iterated logarithm for weighted sums of independent random variables," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 247-254, April.
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    Cited by:

    1. A. Guillou & P. Naveau & J. Diebolt & P. Ribereau, 2009. "Return level bounds for discrete and continuous random variables," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 584-604, November.
    2. Guglielmo D'Amico & Riccardo De Blasis & Philippe Regnault, 2020. "Confidence sets for dynamic poverty indexes," Papers 2006.06595, arXiv.org.
    3. Fontanari, Andrea & Taleb, Nassim Nicholas & Cirillo, Pasquale, 2018. "Gini estimation under infinite variance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 256-269.
    4. Andrea Fontanari & Nassim Nicholas Taleb & Pasquale Cirillo, 2017. "Gini estimation under infinite variance," Papers 1707.01370, arXiv.org, revised Dec 2017.
    5. Boistard Hélène, 2007. "Large deviations for L-statistics," Statistics & Risk Modeling, De Gruyter, vol. 25(2/2007), pages 1-37, April.

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