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Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions

  • Haiwu Huang EMAIL logo , Hongguo Zeng and Yanqin Fan
Published/Copyright: August 4, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 4

ABSTRACT

In this work, suppose that {X n ; n ≥ 1}is a sequence of asymptotically negatively associated random variables and {a ni ; 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers such that ∑i=1n|ani|q=O(n) for some q > max {αp−1α−1/2,2} with αp > 1 and α>12 . Let l (x) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form

∑n=1∞nαp−2l(n)P(max1≤j≤n|∑i=1janiXi|>εnα)<∞ for all ε > 0.

As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained.

2020 Mathematics Subject Classification: Primary 60F15
Keywords: asymptotically negatively associated random variables; complete convergence; strong law of large numbers; weighted sums

(Communicated by Gejza Wimmer)

Funding statement: This paper is supported by Guangxi Special Project of Science and Technology Base and Talent Development (Guike AD23026016) and the Doctor and the Professor Natural Science Foundation of Guilin University of Aerospace Technology (KX202103701).

Acknowledgement

The authors are most grateful to the Editor Professor Gejza Wimmer and two anonymous referees for carefully reading the manuscript and for offering some valuable suggestions and comments, which greatly helped in improving an earlier version of this paper.

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Received: 2022-04-25
Accepted: 2022-10-26
Published Online: 2023-08-04

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0076
Keywords for this article
asymptotically negatively associated random variables; complete convergence; strong law of large numbers; weighted sums

Articles in the same Issue

  1. A Note on Special Subsets of the Rudin-Frolík Order for Regulars
  2. The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
  3. The Deranged Bell Numbers
  4. On Index and Monogenity of Certain Number Fields Defined by Trinomials
  5. The k-Generalized Lucas Numbers Close to a Power of 2
  6. Shifted Power of a Polynomial with Integral Roots
  7. Further Insights into the Mysteries of the Values of Zeta Functions at Integers
  8. Memoryless Properties on Time Scales
  9. A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
  10. Besov and Triebel-Lizorkin Capacity in Metric Spaces
  11. Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
  12. An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
  13. Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
  14. A New Series Space Derived by Absolute Generalized Nörlund Means
  15. Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
  16. The Uniform Effros Property and Local Homogeneity
  17. Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
  18. The Unit-Gompertz Quantile Regression Model for the Bounded Responses
  19. An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
  20. Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
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