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Statistica Sinica 7(1997), 923-928



Zhidong Bai, Philip E. Cheng* and Cun-Hui Zhang

National Sun Yat-Sen University, *Academia Sinica
*National Dong Hwa University and Rutgers University

Abstract: A strong law is established for linear statistics that are weighted sums of a random sample. Using an observation of Cheng (1995a) about the Bernstein and Kolmogorov inequalities, we present an extension to the Hardy-Littlewood strong law under certain moment conditions on the weights and the distribution. As a byproduct, the Marcinkiewicz-Zygmund strong law and the law of the iterated logarithm are obtained for linear statistics with slowly varying weights. The results are applicable to some commonly used linear statistics, especially a family of linear order statistics and some nonparametric regression estimators which motivate the study.

Key words and phrases: Hardy-Littlewood strong law, linear statistics, Marcinkiewicz-Zygmund strong law, weighted sums.

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