An extension of the Hardy-Littlewood strong law. (English) Zbl 1067.60501

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.


60F15 Strong limit theorems