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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| 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. |
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| AbstractList | 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. |
| Author | Bui Khanh, Hang Le Van, Dung Tran Manh, Cuong Ta Cong, Son |
| Author_xml | – sequence: 1 givenname: Son surname: Ta Cong fullname: Ta Cong, Son organization: VNU University of Science, Vietnam National University – sequence: 2 givenname: Cuong surname: Tran Manh fullname: Tran Manh, Cuong email: cuongtm@vnu.edu.vn organization: VNU University of Science, Vietnam National University – sequence: 3 givenname: Hang surname: Bui Khanh fullname: Bui Khanh, Hang organization: VNU University of Science, Vietnam National University – sequence: 4 givenname: Dung surname: Le Van fullname: Le Van, Dung organization: The University of Da Nang - University of Science and Education |
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| CitedBy_id | crossref_primary_10_1016_j_spl_2025_110434 crossref_primary_10_1007_s00362_025_01666_1 crossref_primary_10_3390_math12010149 crossref_primary_10_1007_s10986_025_09687_x |
| Cites_doi | 10.1080/02331888.2019.1615912 10.1016/j.spl.2014.04.028 10.1080/02331888.2018.1436548 10.1023/A:1007849609234 10.1214/aos/1032526952 10.1016/S0252-9602(06)60028-3 10.1007/s10959-019-00973-2 10.1214/aos/1176346079 10.1073/pnas.33.2.25 10.1007/s10114-010-8177-5 10.1080/03610928108828102 10.1090/S0002-9947-1965-0198524-1 10.1007/978-1-4614-4708-5 10.1214/aop/1046294315 10.1080/00949655.2021.1961770 10.1007/s13398-017-0399-2 10.1016/j.camwa.2011.10.018 10.1007/s00362-009-0241-9 |
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| Keywords | 62G05 Complete convergence Simple linear regression model with random design Non parametric regression model with random design Randomly weighted sum 62G20 60F15 |
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| Title | 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 |
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