Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEAK LAWS OF LARGE NUMBERS FOR ARRAYS UNDER A CONDITION OF UNIFORM INTEGRABILITY

  • Sung, Soo-Hak (Department of Applied Mathematics Pai Chai University) ;
  • Lisawadi, Supranee (Department of Mathematics and Statisitcs University of Regina) ;
  • Volodin, Andrei (Department of Mathematics and Statisitcs University of Regina)
  • Published : 2008.01.31
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Abstract

For an array of dependent random variables satisfying a new notion of uniform integrability, weak laws of large numbers are obtained. Our results extend and sharpen the known results in the literature.

Keywords

References

  1. T. K. Chandra, Uniform integrability in the Cesaro sense and the weak law of large numbers, Sankhya Ser. A 51 (1989), no. 3, 309-317
  2. T. K. Chandra and A. Goswami, Cesaro ${\alpha}$-integrability and laws of large numbers I, J. Theoret. Probab. 16 (2003), no. 3, 655-669 https://doi.org/10.1023/A:1025620532404
  3. N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 1, 119-122 https://doi.org/10.1007/BF01013465
  4. A. Gut, The weak law of large numbers for arrays, Statist. Probab. Lett. 14 (1992), no. 1, 49-52 https://doi.org/10.1016/0167-7152(92)90209-N
  5. D. H. Hong and K. S. Oh, On the weak law of large numbers for arrays, Statist. Probab. Lett. 22 (1995), no. 1, 55-57 https://doi.org/10.1016/0167-7152(94)00047-C
  6. K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295 https://doi.org/10.1214/aos/1176346079
  7. V. Kruglov, Growth of sums of pairwise-independent random variables with infinite means, Teor. Veroyatn. Primen. 51 (2006), no. 2, 382-385 https://doi.org/10.4213/tvp60
  8. D. Landers and L. Rogge, Laws of large numbers for pairwise independent uniformly integrable random variables, Math. Nachr. 130 (1987), 189-192 https://doi.org/10.1002/mana.19871300117
  9. D. Li, A. Rosalsky, and A. Volodin, On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 2, 281-305
  10. M. Ordonez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (1994), no. 2, 121-132
  11. M. Ordonez Cabrera and A. Volodin, 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 (2005), no. 2, 644-658 https://doi.org/10.1016/j.jmaa.2004.12.025
  12. Q. M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), no. 2, 343-356 https://doi.org/10.1023/A:1007849609234
  13. S. H. Sung, Weak law of large numbers for arrays of random variables, Statist. Probab. Lett. 42 (1999), no. 3, 293-298 https://doi.org/10.1016/S0167-7152(98)00219-3
  14. S. H. Sung, T. C. Hu, and A. Volodin, On the weak laws for arrays of random variables, Statist. Probab. Lett. 72 (2005), no. 4, 291-298 https://doi.org/10.1016/j.spl.2004.12.019

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