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Zbl 1067.60501
Bai, Zhidong; Cheng, Philip E.; Zhang, Cun-Hui
An extension of the Hardy-Littlewood strong law. (English)
[J] Stat. Sin. 7, No. 4, 923-928 (1997). ISSN 1017-0405

Summary: A strong law is established for linear statistics that are weighted sums of a random sample. Using an observation of Cheng (1995) 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.

MSC 2000:: *60F15 Strong limit theorems

Keywords: linear statistics; Marcinkiewicz-Zygmund strong law; weighted sums

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Zentralblatt MATH Berlin [Germany]