Complete Convergence for Weighted Sums of Pairwise Negative Quadrant Dependent Random Variables
Main Article Content
Abstract
In this work, we developed Jajte’s technique of the strong law of large numbers to the complete convergence for randomly weighted sums of pairwise negative quadrant dependent random variables.
Keywords:
Complete convergence, Randomly weighted sum, Negative quadrant dependence.
References
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[9] T. C. Son, T. M. Cuong, L. V. Dung, T.V. Chien, On the Convergence for Weighted Sums of Hilbert-Valued Coordinatewise Pairwise NQD Random Variables and Its Application, Communications in Statistics-Theory and Methods, Vol. 49, 2022, pp. 2770-2786.
[10] N. T. T. Hien, L. V. Thanh, V. Van, On the Negative Dependence in Hilbert Spaces with Applications, Applications of Mathematics, Vol. 64, 2019, pp. 45-59.
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[12] P. Chen, T. Zhang, S. H. Sung, Strong Laws for Randomly Weighted Sums of Random Variables and Applications in The Bootstrap and Random Design Regression, Statistica Sinica, Vol. 29, 2019, pp. 1739-1749.
[13] X. Deng, X. J. Wang, S. Hu, M. Hu, A General Result on Complete Convergence for Weighted Sums of Linear Processes and Its Statistical Applications, Statistics, Vol. 53, 2019, pp. 903-920.
[14] E. L. Lehmann, Somes Concepts of Dependence, Ann. Math. Stat., Vol. 37, 1966, pp. 1137-1153.
[15] K. Alam, K. M. L. Saxena, Positive Dependence in Multivariate Distributions, Commun Stat Theory Methods, Vol. 10, 1981, pp. 1183-1196.
[16] N. T. T. Hien, L. V. Thanh, On the Weak Laws of Large Numbers for Sums of Negatively Associated Random Vectors in Hilbert Spaces, Stat. Probab. Lett., Vol. 107, 2015, pp. 236-245.
[17] D. Li, A. Rosalsky, A. Volodin, On the Strong Law of Large Numbers for Sequences of Pairwise Negative Quadrant Dependent Random Variables. Bulletin of the Institute of Mathematics Academia Sinica, New Series, Vol. 1, 2006, pp. 281-305.
[18] A. Gut, Probability: A Graduate Course, Second Edition, Springer New York, NY, 2013.
[19] N. Bingham, C. Goldie, J. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.
Vol. 33, 1947, pp. 25-31.
[2] L. E. Baum, M. Katz, Convergence Rates in the Law of Large Numbers, Trans. Amer. Math. Soc., Vol. 120, 1965, pp. 108-123.
[3] K. J. Dev, F. Proschan, Negative Association of Random Variables with Applications, Ann. Stat., Vol. 11, 1983, pp. 286-295.
[4] S.H. Sung, On the Strong Law of Large Numbers for Weighted Sums of Random Variables, Computers and Mathematics with Applications, Vol. 62, 2011, pp. 4277-4287.
[5] X. Wang, X. Li, S. Hu, W. Yang, Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables, Stochastic Analysis and Applications, Vol. 29, 2011, pp. 1-14.
[6] D. H. Qiu, K. C. Chang, R. G. Antoninic, A. Volodin, On the Strong Rates of Convergence for Arrays of Rowwise Negatively Dependent Random Variables, Stoch. Anal. Appl, Vol. 29, 2011, pp. 375-385.
[7] Q. Y. Wu, Convergence Properties of Pairwise NQD Random Sequences, Acta Math. Sin. Chin. Ser., Vol. 45, 2002, pp. 617-624.
[8] L. V. Dung, T. C. Son, T. M. Cuong, Weak Laws of Large Numbers for Weighted Coordinatewise Pairwise Random Vectors in Hilbert Spaces, J. Korean Math. Soc., Vol. 56, 2019, pp. 457-473.
[9] T. C. Son, T. M. Cuong, L. V. Dung, T.V. Chien, On the Convergence for Weighted Sums of Hilbert-Valued Coordinatewise Pairwise NQD Random Variables and Its Application, Communications in Statistics-Theory and Methods, Vol. 49, 2022, pp. 2770-2786.
[10] N. T. T. Hien, L. V. Thanh, V. Van, On the Negative Dependence in Hilbert Spaces with Applications, Applications of Mathematics, Vol. 64, 2019, pp. 45-59.
[11] R. Jajte, On the Strong Law of Large Numbers, The Annals of Probability, Vol. 31, 2003, pp. 409-412.
[12] P. Chen, T. Zhang, S. H. Sung, Strong Laws for Randomly Weighted Sums of Random Variables and Applications in The Bootstrap and Random Design Regression, Statistica Sinica, Vol. 29, 2019, pp. 1739-1749.
[13] X. Deng, X. J. Wang, S. Hu, M. Hu, A General Result on Complete Convergence for Weighted Sums of Linear Processes and Its Statistical Applications, Statistics, Vol. 53, 2019, pp. 903-920.
[14] E. L. Lehmann, Somes Concepts of Dependence, Ann. Math. Stat., Vol. 37, 1966, pp. 1137-1153.
[15] K. Alam, K. M. L. Saxena, Positive Dependence in Multivariate Distributions, Commun Stat Theory Methods, Vol. 10, 1981, pp. 1183-1196.
[16] N. T. T. Hien, L. V. Thanh, On the Weak Laws of Large Numbers for Sums of Negatively Associated Random Vectors in Hilbert Spaces, Stat. Probab. Lett., Vol. 107, 2015, pp. 236-245.
[17] D. Li, A. Rosalsky, A. Volodin, On the Strong Law of Large Numbers for Sequences of Pairwise Negative Quadrant Dependent Random Variables. Bulletin of the Institute of Mathematics Academia Sinica, New Series, Vol. 1, 2006, pp. 281-305.
[18] A. Gut, Probability: A Graduate Course, Second Edition, Springer New York, NY, 2013.
[19] N. Bingham, C. Goldie, J. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.