Abstract
Some convergence results in mean of order p for arrays of row-wise extended negatively dependent random variables are presented under asymptotic integrability conditions. A Rosenthal type inequality for these dependent structures is also announced playing a central role in our approach to this issue. As consequence, well-known results about convergence in p-mean for random variables will be extended.
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Adler A., Rosalsky A., Volodin A.I.: A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces. Statist. Probab. Lett., 32, 167–174 (1997)
Cabrera M.O., 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, 644–658 (2005)
Chen Y., Chen A., Ng K.W.: The strong law of large numbers for extended negatively dependent random variables. J. Appl. Probab., 47, 908–922 (2010)
Chen Y., Wang L., Wang Y.: Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl., 401, 114–129 (2013)
Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 3rd ed., Springer-Verlag (New York, 1997).
P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events, Springer-Verlag (Berlin–Heidelberg, 1997).
Gut A.: Complete convergence for arrays. Period. Math. Hungar., 25, 51–75 (1992)
Gut A.: The weak law of large numbers for arrays. Statist. Probab. Lett., 14, 49–52 (1992)
Lehmann E.L.: Some concepts of dependence. Ann. Math. Statist., 37, 1137–1153 (1966)
Lita da Silva J.: Almost sure convergence for weighted sums of extended negatively dependent random variables. Acta Math. Hungar., 146, 56–70 (2015)
Lita da Silva J.: Limiting behavior for arrays of row-wise upper extended negatively dependent random variables. Acta Math. Hungar., 148, 481–492 (2016)
V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Oxford Studies in Probability, Vol. 4, Clarendon Press (Oxford, 1995).
Pyke R., Pyke R.: On convergence in r-mean of normalized partial sums. Ann. Math. Statist., 39, 379–381 (1968)
A. Shen, Probability inequalities for END sequence and their applications, J. Inequal. Appl., 98 (2011), pp. 12.
Sung S.H.: Convergence in r-mean of weighted sums of NQD random variables. Appl. Math. Lett., 26, 18–24 (2013)
Wu Y., Guan M.: Convergence properties of the partial sums for sequences of END random variables. J. Korean Math. Soc., 49, 1097–1110 (2012)
Y. Wu, M. Song and C. Wang, Complete moment convergence and mean convergence for arrays of rowwise extended negatively dependent random variables. The Scientific World Journal, Article ID 478612, Vol. 2014, pp. 7.
Yuan D., Tao B.: Mean convergence theorems for weighted sums of arrays of residually h-integrable random variables concerning the weights under dependence assumptions. Acta Appl. Math., 103, 221–234 (2008)
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J. Lita da Silva This work is a contribution to the Project UID/GEO/04035/2013, funded by FCT — Fundação para a Ciência e a Tecnologia, Portugal.
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da Silva, J.L. Convergence in p-mean for arrays of row-wise extended negatively dependent random variables. Acta Math. Hungar. 150, 346–362 (2016). https://doi.org/10.1007/s10474-016-0645-7
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DOI: https://doi.org/10.1007/s10474-016-0645-7