Strong Laws of Large Numbers for Weighted Sums of Hilbert-valued Coordinatewise PNQD Random Variables with Applications
Main Article Content
Abstract
The aim of this work is to investigate results on almost sure convergence of weighted sums of coordinatewise pairwise negatively quadrant dependent random variables taking values in Hilbert spaces. As an application, the almost sure convergence of degenerate von Mises-statistics is investigated.
Keywords:
Negative quadrant dependence, Hilbert spaces, Weighted sums, Strong laws of large numbers.
References
[1] E. L. Lehmann, Somes Concepts of Dependence, Ann. Math. Stat, Vol. 37, No. 5, 1966, pp. 1137-1153, http://doi.org/10.1214/aoms/1177699260.
[2] N. Ebrahimi, M. Ghosh, Multivariate Negative Dependence, Commun. Stat., Theory Methods, Vol. 10, 1981,
pp. 307-337, https://doi.org/10.1080/03610928108828041.
[3] L. V. Dung, T. C. Son, T. M. Cuong, Weak Laws of Large Numbers for Weighted Coordinatewise Pairwise NQD Random Vectors in Hilbert Spaces, Journal of the Korean Mathematical Society, Vol. 56, No. 2, 2019,
pp. 457-473, https://doi.org/10.4134/JKMS.j180217.
[4] N. T. T. Hien, L. V. Thanh, V. T. H. Van, on the Negative Dependence in Hilbert Spaces with Applications, Appl. Math., Vol. 64, No. 1, 2019, pp. 45-59, https://doi.org/10.21136/AM.2018.0060-18.
[5] T. C. Son, T. M. Cuong, L. V. Dung, V. T. Chien, on the Convergence for Weighted Sums of Hilbert-valued Coordinatewise Pairwise NQD Random Variables and Its application, Communications in Statistics - Theory and Methods, 2022, https://doi.org/10.1080/03610926.2022.2062604.
[6] N. H. Bingham , C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Series 27, Cambridge University Press, 1987.
[7] F. C. Cirstea, V. Radulescu, Nonlinear Problems with Boundary Blow-up: a Karamata Regular Variation Theory Approach, Asymptotic Analysis, Vol. 46, No. 3-4, 2006, pp. 275-298.
[8] H. Sun, Mercer Theorem for RKHS on Noncompact Sets, J. Compl., Vol. 21, 2005, pp. 337-349, https://doi.org/10.1016/j.jco.2004.09.002.
[9] H. Dehling, O. S. Sharipov, M. Wendler, Bootstrap for Dependent Hilbert Space-Valued Random Variables with Application to Von Mises Statistics, J. Multivariate Anal., Vol. 133, 2015, pp. 200-215, https://doi.org/10.1016/j.jmva.2014.09.011.
[10] T. C. Son, L. V. Dung, D. T. Dat, T. T. Trang, Generalized Marcinkiewicz laws for Weighted Dependent Random Vectors in Hilbert spaces, Theory of Probability and its Applications, Vol. 67, No. 3, 2022, pp. 541-562, https://doi.org/10.4213/tvp5436.
[11] X. Wang, S. Hu, A. I. Volodin, Strong Limit Theorems for Weighted Sums of Nod Sequence and Exponential Inequalities, Bull. Korean Math. Soc., Vol. 48, No. 5, 2011, pp. 923-938, http://dx.doi.org/10.4134/BKMS.2011.48.5.923.
[12] A. Shen, H. Zhu, R. Wu, Y. Zhang, Complete Convergence for Weighted Sums of LNQD Random Variables, Stochastics An International Journal of Probability and Stochastic Processes, Vol. 87, No. 1, 2015, pp. 160-169, https://doi.org/10.1080/17442508.2014.931959.
[13] T. C. Son, T. M. Cuong, L. V. Dung, on the Almost Sure Convergence for Sums of Negatively Superadditive Dependent Random Vectors in Hilbert Spaces and Its Application, Commun. Statist. Theor. Methods, Vol. 49, 2020, pp. 2770-2786, https://doi.org/10.1080/03610926.2019.1584304.
[2] N. Ebrahimi, M. Ghosh, Multivariate Negative Dependence, Commun. Stat., Theory Methods, Vol. 10, 1981,
pp. 307-337, https://doi.org/10.1080/03610928108828041.
[3] L. V. Dung, T. C. Son, T. M. Cuong, Weak Laws of Large Numbers for Weighted Coordinatewise Pairwise NQD Random Vectors in Hilbert Spaces, Journal of the Korean Mathematical Society, Vol. 56, No. 2, 2019,
pp. 457-473, https://doi.org/10.4134/JKMS.j180217.
[4] N. T. T. Hien, L. V. Thanh, V. T. H. Van, on the Negative Dependence in Hilbert Spaces with Applications, Appl. Math., Vol. 64, No. 1, 2019, pp. 45-59, https://doi.org/10.21136/AM.2018.0060-18.
[5] T. C. Son, T. M. Cuong, L. V. Dung, V. T. Chien, on the Convergence for Weighted Sums of Hilbert-valued Coordinatewise Pairwise NQD Random Variables and Its application, Communications in Statistics - Theory and Methods, 2022, https://doi.org/10.1080/03610926.2022.2062604.
[6] N. H. Bingham , C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Series 27, Cambridge University Press, 1987.
[7] F. C. Cirstea, V. Radulescu, Nonlinear Problems with Boundary Blow-up: a Karamata Regular Variation Theory Approach, Asymptotic Analysis, Vol. 46, No. 3-4, 2006, pp. 275-298.
[8] H. Sun, Mercer Theorem for RKHS on Noncompact Sets, J. Compl., Vol. 21, 2005, pp. 337-349, https://doi.org/10.1016/j.jco.2004.09.002.
[9] H. Dehling, O. S. Sharipov, M. Wendler, Bootstrap for Dependent Hilbert Space-Valued Random Variables with Application to Von Mises Statistics, J. Multivariate Anal., Vol. 133, 2015, pp. 200-215, https://doi.org/10.1016/j.jmva.2014.09.011.
[10] T. C. Son, L. V. Dung, D. T. Dat, T. T. Trang, Generalized Marcinkiewicz laws for Weighted Dependent Random Vectors in Hilbert spaces, Theory of Probability and its Applications, Vol. 67, No. 3, 2022, pp. 541-562, https://doi.org/10.4213/tvp5436.
[11] X. Wang, S. Hu, A. I. Volodin, Strong Limit Theorems for Weighted Sums of Nod Sequence and Exponential Inequalities, Bull. Korean Math. Soc., Vol. 48, No. 5, 2011, pp. 923-938, http://dx.doi.org/10.4134/BKMS.2011.48.5.923.
[12] A. Shen, H. Zhu, R. Wu, Y. Zhang, Complete Convergence for Weighted Sums of LNQD Random Variables, Stochastics An International Journal of Probability and Stochastic Processes, Vol. 87, No. 1, 2015, pp. 160-169, https://doi.org/10.1080/17442508.2014.931959.
[13] T. C. Son, T. M. Cuong, L. V. Dung, on the Almost Sure Convergence for Sums of Negatively Superadditive Dependent Random Vectors in Hilbert Spaces and Its Application, Commun. Statist. Theor. Methods, Vol. 49, 2020, pp. 2770-2786, https://doi.org/10.1080/03610926.2019.1584304.