On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

References

[1] Adler, A.—Ordóñez Cabrera, M.—Rosalsky, A.—Volodin, A. I.: Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces, Bull. Inst. Math. Acad. Sin. (N.S.) 27(3) (1999), 187–212.Search in Google Scholar

[2] Anh, V. T. N.—Hien, N. T. T.—Thanh, L. V.—Van, V. T. H.: The Marcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theoret. Probab. 34(1) (2021), 331–348.Search in Google Scholar

[3] Asadian, N.—Fakour, V.—Bozorgnia, A.: Rosenthals type inequalities for negatively orthant dependent random variables, J. Iran. Stat. Soc. (JIRSS) 5 (2006), 69–75.Search in Google Scholar

[4] von Bahr, B.—Esseen, C. G.: Inequalities for the r th absolute moment of a sum of random variables, 1 ≤ r ≤ 2, Ann. Math. Statist. 36(1) (1965), 299–303.Search in Google Scholar

[5] Bose, A.—Chandra, T. K.: Cesáro uniform integrability and L_p convergence, Sankhya A 3 (1993), 12–28.Search in Google Scholar

[6] Bryc, W.—Smoleński, W.: Moment conditions for almost sure convergence of weakly correlated random variables, Proc. Amer. Math. Soc. 119(2) (1993), 629–635.Search in Google Scholar

[7] Chandra, T. K.: Uniform integrability in the Cesàro sense and the weak law of large numbers, Sankhya A 51(3) (1989), 309–317.Search in Google Scholar

[8] Chandra, T. K.—Goswami, A.: Cesaro uniform integrability and the strong law of large numbers, Sankhya A 51 (1992), 215–231.Search in Google Scholar

[9] Chatterji, S. D.: An L_p-convergence theorem, Ann. Math. Statist. 40(3) (1969), 1068–1070.Search in Google Scholar

[10] Chen, L. H. Y.—Raič, M.—Thành, L. V.: On the error bound in the normal approximation for Jack measures, Bernoulli 27(1) (2021), 442–468.Search in Google Scholar

[11] Chen, P.—Bai, P.—Sung, S. H.: The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419(2) (2014), 1290–1302.Search in Google Scholar

[12] Chen, P.—Sung, S. H.: Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848.Search in Google Scholar

[13] Fazekas, I.—Pecsora, S.: General Bahr–Esseen inequalities and their applications, J. Inequal. Appl. 2017(1) (2017), 1–16.Search in Google Scholar

[14] Hien, N. T. T.—Thanh, L. V.—Van, V. T. H.: On the negative dependence in Hilbert spaces with applications, Appl. Math. 64(1) (2019), 45–59.Search in Google Scholar

[15] Hong, D. H.—Hwang, S. Y.: Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables, Int. J. Math. Math. Sci. 22(1) (1999), 171–177.Search in Google Scholar

[16] Johnson, W. B.—Schechtman, G.—Zinn, J.: Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13(1) (1985), 234–253.Search in Google Scholar

[17] Ordóñez Cabrera, M.: Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (1994), 121–132.Search in Google Scholar

[18] Ordóñez Cabrera, M.—Rosalsky, A.—Ünver, M.—Volodin, A.: A new type of compact uniform integrability with application to degenerate mean convergence of weighted sums of Banach space valued random elements, J. Math. Anal. Appl. 487 (2020), Art. ID 123975.Search in Google Scholar

[19] Ordóñez Cabrera, M.—Volodin, A. I.: Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305(2) (2005), 644–658.Search in Google Scholar

[20] Pinelis, I.: On the von Bahr–Esseen inequality, preprint (2010); https://arxiv.org/abs/1008.5350.Search in Google Scholar

[21] Pinelis, I.: Best possible bounds of the von Bahr–Esseen type, Ann. Funct. Anal. 6(4) (2015), 1–29.Search in Google Scholar

[22] Rosalsky, A.—Thành, L. V.: A note on the stochastic domination condition and uniform integrability with applications to the strong law of large numbers, Statist. Probab. Lett. 178 (2021), Art. ID 109181.Search in Google Scholar

[23] Rosenthal, H. P.: On the subspaces of L^p (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8(3) (1970), 273–303.Search in Google Scholar

[24] Sung, S. H.: Convergence in r-mean of weighted sums of NQD random variables, Appl. Math. Lett. 26(1) (2013), 18–24.Search in Google Scholar

[25] Taylor, R. L.: Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces. Lecture Notes in Math., Vol. 672, Springer–Verlag, Berlin, 1978.Search in Google Scholar

[26] Thành, L. V.: On the L^p-convergence for multidimensional arrays of random variables, Int. J. Math. Math. Sci. 27(8) (2005), 1317–1320.Search in Google Scholar

[27] Thành, L. V.: Mean convergence theorems and weak laws of large numbers for double arrays of random variables, Int. J. Stoch. Anal. 2006 (2006), Art. ID 049561.Search in Google Scholar