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.