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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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Son Ta Cong & Cuong Tran Manh & Hang Bui Khanh & Dung Le Van, 2024. "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," Statistical Papers, Springer, vol. 65(3), pages 1869-1900, May.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:3:d:10.1007_s00362-023-01483-4
    DOI: 10.1007/s00362-023-01483-4
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