Abstract
This paper introduces the concept of double arrays of blockwise M-pairwise negatively dependent random variables and establishes \(L^1\) convergence for double arrays of blockwise M-pairwise negatively dependent random variables. Our proof is completely different from those of the aforementioned papers.
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References
Lehmann, E.: Some concepts of dependence. Ann. Math. Stat. 37(5), 1137–1153 (1966)
Anh, V.T.V., Anh, N.T.N., Hien, N.T.T., Tu, N.N.: Convergence in mean for double arrays of \(m\)-pairwise negatively dependent random variables. Bulletin Malays. Math. Sci. Soc. 45(4), 1507–1520 (2022)
Stadtmüller, U., Thành, L.V.: On the strong limit theorems for double arrays of blockwise \(M\)-dependent random variables. Acta Math. Sinica Engl. Series 27(10), 1923–1934 (2011)
Huan, N.V., Quang, N.V., Volodin, A.I.: Strong laws for blockwise martingale difference arrays in banach spaces. Lobachevskii J. Math. 31(4), 326–335 (2010)
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, 109181 (2021)
Czerebak-Mrozowicz, E.B., Klesov, O.I., Rychlik, Z.: Marcinkiewicz-type strong law of large numbers for pairwise independent random fields. Probab. Math. Stat. 22(1), 127–139 (2002)
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), 171–177 (1999)
Hong, D.H., Volodin, A.I.: Marcinkiewicz-type law of large numbers for double arrays. J. Korean Math. Soc. 36(6), 1133–1143 (1999)
Rosalsky, A., Thành, L.V.: Weak laws of large numbers for double sums of independent random elements in Rademacher type \(p\) and stable type \(p\) Banach spaces. Nonlinear Anal. Theory Methods Appl. 71(12), 1065–1074 (2009)
Thành, L.V.: On the \(L_p\)-convergence for multidimensional arrays of random variables. Int. J. Math. Math. Sci. 2005(8), 1317–1320 (2005)
Klesov, O.: Limit Theorems for Multi-indexed Sums of Random Variables vol. 71, (2014)
Rosalsky, A., Thành, L.V.: On almost sure and mean convergence of normed double sums of Banach space valued random elements. Stoch. Anal. Appl. 25(4), 895–911 (2007)
Móricz, F.: The Kronecker lemmas for multiple series and some application. Acta Mathematica Academiae Scientiarum Hungarica 37(1), 39–50 (1981)
Anh, V.T.N.: On the strong laws of large numbers for sequences of blockwise pairwise and coordinatewise negatively dependent random vectors in hilbert spaces. Lobachevskii J. Math. 42(13), 3077–3087 (2021)
Hien, N.T.T., Thành, L.V., Van, V.T.H.: On the negative dependence in Hilbert spaces with applications. Appl. Math. 64(1), 45–59 (2019)
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Communicated by See Keong Lee.
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Vu, T.N.A. On the \( L^1 \)-Convergence for Double Arrays of Blockwise M-Pairwise Negatively Dependent Random Variables. Bull. Malays. Math. Sci. Soc. 46, 172 (2023). https://doi.org/10.1007/s40840-023-01558-3
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DOI: https://doi.org/10.1007/s40840-023-01558-3