On the Baum–Katz theorem for randomly weighted sums of negatively associated random variables with general normalizing sequences and applications in some random design regression models

Son Ta Cong, Cuong Tran Manh (), Hang Bui Khanh and Dung Le Van
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Son Ta Cong: Vietnam National University
Cuong Tran Manh: Vietnam National University
Hang Bui Khanh: Vietnam National University
Dung Le Van: The University of Da Nang - University of Science and Education

Statistical Papers, 2024, vol. 65, issue 3, No 26, 1869-1900

Abstract: Abstract In this paper, we develop Jajte’s technique, which is used in the proof of strong laws of large numbers, to prove complete convergence for randomly weighted sums of negatively associated random variables. Based on a general normalizing function that satisfies some specific conditions, we give some general results on complete convergence for randomly weighted sums of random variables. The Baum–Katz theorem for randomly weighted sums with general normalizing sequences is also presented. Our results have an interesting connection with the theory of regularly varying functions. These results are applied to simple linear regression models as well as nonparametric regression models with random design. Furthermore, simulations to study the numerical performance of the consistency for nearest neighbor weight function estimators in nonparametric regression and least-squares estimators in a simple linear regression with random design are given.

Keywords: Complete convergence; Simple linear regression model with random design; Non parametric regression model with random design; Randomly weighted sum; 60F15; 62G05; 62G20 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00362-023-01483-4

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