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

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 g...

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Bibliographic Details
Published in:Statistical papers (Berlin, Germany) Germany), 2024-05, Vol.65 (3), p.1869-1900
Main Authors: Ta Cong, Son, Tran Manh, Cuong, Bui Khanh, Hang, Le Van, Dung
Format: Article
Language:English
Subjects:
Convergence
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Estimators
Finance
Insurance
Management
Mathematics and Statistics
Operations Research/Decision Theory
Probability Theory and Stochastic Processes
Random variables
Regression analysis
Regression models
Regular Article
Statistics
Statistics for Business
Sums
Theorems
Weighting functions
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Summary: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.
ISSN:0932-5026
1613-9798
DOI:10.1007/s00362-023-01483-4