Conditional mean convergence theorems of conditionally dependent random variables under conditions of integrability

Abstract

We give the conditionally residual h-integrability with exponent r for an array of random variables and establish the conditional mean convergence of conditionally negatively quadrant dependent and conditionally negative associated random variables under this integrability. These results generalize and improve the known ones.

Keywords

Conditional negatively quadrant dependent (NQD) random variable / conditional negatively associated (NA) random variable / conditional mean convergence / conditionally residual h-integrability

References

[1] Chandra T K, Goswami A. Cesàro α-integrability and laws of large numbers I. J Theoret Probab, 2003, 16: 655-669

[2] Chandra T K, Goswami A. Cesàro α-integrability and laws of large numbers II. J Theoret Probab, 2006, 19: 789-816

[3] Chow Y S, Teicher H. Probability Theory: Independence, Interchangeability, Martingales. 2nd ed. New York: Springer-Verlag, 1988

[4] Christofides T C, Hadjikyriakou M. Conditional demimartingales and related results. J Math Anal Appl, 2013, 398: 380-391

[5] Joag-Dev K, Proschan F. Negative association of random variables, with applications. Ann Statist, 1983, 11: 286-295

[6] Lehmann E L. Some concepts of dependence. Ann Math Statist, 1966, 37: 1137-1153

[7] Liu J C, Prakasa Rao B L S. On conditional Borel-Cantelli lemmas for sequences of random variables. J Math Anal Appl, 2013, 399: 156-165

[8] Majerek D, Nowak W, Zieba W. Conditional strong law of large number. Int J Pure Appl Math, 2005, 20: 143-157

[9] Ordóñez Cabrera M, Rosalsky A, Volodin A. Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables. Test, 2012, 21: 369-385

[10] Ordóñez Cabrera M, Volodin A. 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, 2005, 305: 644-658

[11] Prakasa Rao B L S. Conditional independence, conditional mixing and conditional association. Ann Inst Statist Math, 2009, 61: 441-460

[12] Prakasa Rao B L S. Upper and lower bounds for probabilities in the conditional Borel-Cantelli lemma. Stoch Anal Appl, 2010, 28: 144-156

[13] Roussas G G. On conditional independence, mixing, and association. Stoch Anal Appl, 2008, 26: 1274-1309

[14] Shen A T, Wu R C, Chen Y, Zhou Y. Conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability. J Inequal Appl, 2013, article 122

[15] Sung S H. Weak law of large numbers for arrays of random variables. Statist Probab Lett, 1999, 42: 293-298

[16] Sung S H, Lisawadi S, Volodin A. Weak laws of large numbers for arrays under a condition of uniform integrability. J Korean Math Soc, 2008, 45: 289-300

[17] Wang X H, Hu S H. Weak laws of large numbers for arrays of dependent random variables. Stochastics, 2014, 86: 759-775

[18] Wang X H, Wang X J. Some inequalities for conditional demimartingales and conditional N-demimartingales. Statist Probab Lett, 2013, 83: 700-709

[19] Yuan D M, An J, Wu X S. Conditional limit theorems for conditionally negatively associated variables. Monatsh Math, 2010, 161: 449-473

[20] Yuan D M, Lei L. Some conditional results for conditionally strong mixing sequences of random variables. Sci China Ser A, 2013, 56: 845-859

[21] Yuan DM, 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, 2008, 103: 221-234

[22] Yuan D M, Xie Y. Conditional limit theorems for conditionally linearly negative quadrant dependent random variables. Monatsh Math, 2012, 166: 281-299

[23] Yuan D M, Yang Y K. Conditional versions of limit theorems for conditionally associated random variables. J Math Anal Appl, 2011, 376: 282-293

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Higher Education Press and Springer-Verlag Berlin Heidelberg