Weak Laws of Large Numbers for Negatively Superadditive Dependent Random Vectors in Hilbert Spaces
Main Article Content
Abstract
Let $\{X_{n}, {n}\in \mathbb{N}\}$ be a sequence of negatively superadditive dependent random vectors taking values in a real separable Hilbert space. In this paper, we present the weak laws of large numbers for weighted sums (with or without random indices) of $\{X_{n}, {n}\in \mathbb{N}\}$.
Keywords:
Large numbers, negatively superadditive dependent random vectors, Hilbert space.
References
[1] A. Adler, A. Rosalsky, R. L. Taylor, A Weak Law for Normed Weighted Sums of Random Elements in Rademacher Type p Banach Spaces, Journal of Multivariate Analysis, Vol. 37, No. 2, 1991, pp. 259-268, https://doi.org/10.1016/0047-259X(91)90083-E.
[2] T. M. Cuong, T. C. Son, Weak Laws of Large Numbers of Cesaro Summation for Random Arrays, VNU Journal of Science: Mathematics-Physics, Vol. 31, No. 3, 2015, pp. 31-38.
[3] L. V. Dung, T. C. Son, N. T. H. Yen, Weak Laws of Large Numbers for Sequences of Random Variables with Infinite rth Moments, Acta Mathematica Hungarica, Vol. 156, 2018, pp. 408-423, https://doi.org/10.1007/s10474-018-0865-0.
[4] D. H. Hong, M. O. Cabrera, S. H. Sung, A. I. Volodin, On the Weak Law for Randomly Indexed Partial Sums for Arrays of Random Elements in Martingale Type p Banach Spaces, Statistics and Probability Letters, Vol. 46, No. 2, 2000, pp. 177-185, https://doi.org/10.1016/S0167-7152(99)00103-0.
[5] T. C. Son, D. H. Thang, P. V. Thu, Weak Laws of Large Numbers for Fields of Random Variables in Banach Spaces, Journal of Probability and Statistical Science, Vol. 13, No. 2, 2015, pp. 153-165.
[6] N. T. T. Hien, L. V. Thanh, On the Weak Laws of Large Numbers for Sums of Negatively Associated Random Vectors in Hilbert Spaces, Statistics and Probability Letters, Vol. 107, 2015, pp. 236-245, https://doi.org/10.1016/j.spl.2015.08.030.
[7] 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.
[8] T. Z. Hu, Negatively Superadditive Dependence of Random Variables with Applications, Chinese Journal of Applied Probability and Statistics, Vol. 16, No. 2, 2000, pp. 133-144.
[9] 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, Communications in Statistics – Theory and Methods, Vol. 49, No. 11, 2020, pp. 2770-2786, https://doi.org/10.1080/03610926.2019.1584304.
[2] T. M. Cuong, T. C. Son, Weak Laws of Large Numbers of Cesaro Summation for Random Arrays, VNU Journal of Science: Mathematics-Physics, Vol. 31, No. 3, 2015, pp. 31-38.
[3] L. V. Dung, T. C. Son, N. T. H. Yen, Weak Laws of Large Numbers for Sequences of Random Variables with Infinite rth Moments, Acta Mathematica Hungarica, Vol. 156, 2018, pp. 408-423, https://doi.org/10.1007/s10474-018-0865-0.
[4] D. H. Hong, M. O. Cabrera, S. H. Sung, A. I. Volodin, On the Weak Law for Randomly Indexed Partial Sums for Arrays of Random Elements in Martingale Type p Banach Spaces, Statistics and Probability Letters, Vol. 46, No. 2, 2000, pp. 177-185, https://doi.org/10.1016/S0167-7152(99)00103-0.
[5] T. C. Son, D. H. Thang, P. V. Thu, Weak Laws of Large Numbers for Fields of Random Variables in Banach Spaces, Journal of Probability and Statistical Science, Vol. 13, No. 2, 2015, pp. 153-165.
[6] N. T. T. Hien, L. V. Thanh, On the Weak Laws of Large Numbers for Sums of Negatively Associated Random Vectors in Hilbert Spaces, Statistics and Probability Letters, Vol. 107, 2015, pp. 236-245, https://doi.org/10.1016/j.spl.2015.08.030.
[7] 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.
[8] T. Z. Hu, Negatively Superadditive Dependence of Random Variables with Applications, Chinese Journal of Applied Probability and Statistics, Vol. 16, No. 2, 2000, pp. 133-144.
[9] 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, Communications in Statistics – Theory and Methods, Vol. 49, No. 11, 2020, pp. 2770-2786, https://doi.org/10.1080/03610926.2019.1584304.