Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables
Affiliation(s)
Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, Thailand.
Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, Thailand.
ABSTRACT
In this paper, we obtain the strong law of large numbers for a 2-dimensional array of pairwise negatively dependent random variables which are not required to be identically distributed. We found the sufficient conditions of strong law of large numbers for the difference of random variables which independent and identically distributed conditions are regarded. In this study, we consider the limit as which is stronger than the limit as m× n→ ∞ when m, n → ∞ are natural numbers.
Cite this paper
K. Surakamhaeng, N. Chaidee and K. Neammanee, "Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables," Open Journal of Statistics, Vol. 3 No. 1, 2013, pp. 42-46. doi: 10.4236/ojs.2013.31006.
K. Surakamhaeng, N. Chaidee and K. Neammanee, "Strong Law of Large Numbers for a 2-Dimensional Array of Pairwise Negatively Dependent Random Variables," Open Journal of Statistics, Vol. 3 No. 1, 2013, pp. 42-46. doi: 10.4236/ojs.2013.31006.
References
[1] S. Cs¨org?o, K. Tandori and V. Totik, “On the Strong Law of Large Numbers for Pairwise Independent Random Variables,” Acta Mathematica Hungarica, Vol. 42, No. 3- 4, 1983, pp. 319-330. doi:10.1007/BF01956779 [2] N. Etemadi, “An Elementary Proof of the Strong Law of Large Numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 55, No. 1, 1981, pp. 119-122. [3] R. G. Laha and V. K. Rohatgi, “Probability Theory,” John Wiley & Sons, Hoboken, 1979. [4] T. Birkel, “A Note on the Strong Law of Large Numbers for Positively Dependent Random Variables,” Statistics & Probability Letters, Vol. 7, No. 1, 1989, pp. 17-20. doi:10.1016/0167-7152(88)90080-6 [5] H. A. Azarnoosh, “On the Law of Large Numbers for Negatively Dependent Random Variables,” Pakistan Journal of Statistics, Vol. 19, No. 1, 2003, pp. 15-23. [6] H. R. Nili Sani, H. A. Azarnoosh and A. Bozorgnia, “The Strong Law of Large Numbers for Pairwise Negatively Dependent Random Variables,” Iranian Journal of Science & Technology, Vol. 28, No. A2, 2004, pp. 211-217. [7] T. S. Kim, H. Y. Beak and H. Y. Seo, “On Strong Laws of Large Numbers for 2-Dimensional Positively Dependent Random Variables,” Bulletin of the Korean Mathematical Society, Vol. 35, No. 4, 1998, pp. 709-718. [8] T. S. Kim, H. Y. Beak and K. H. Han, “On the Almost Sure Convergence of Weighted Sums of 2-Dimensional Arrays of Positive Dependent Random Variables,” Communications of the Korean Mathematical Society, Vol. 14, No. 4, 1999, pp. 797-804. [9] F. M′oricz, “The Kronecker Lemmas for Multiple Series and Some Applications,” Acta Mathematica Academiae Scientiarum Hungaricae, Vol. 37, No. 1-3, 1981, pp. 39- 50. doi:10.1007/BF01904871 [10] N. Ebrahimi and M. Ghosh, “Multivariate Negative Dependence,” Communications in Statistics—Theory and Methods, Vol. A10, No. 4, 1981, pp. 307-337. [11] K. L. Chung, “A Course in Probability Theory,” Academic Press, London, 2001.
[1] S. Cs¨org?o, K. Tandori and V. Totik, “On the Strong Law of Large Numbers for Pairwise Independent Random Variables,” Acta Mathematica Hungarica, Vol. 42, No. 3- 4, 1983, pp. 319-330. doi:10.1007/BF01956779 [2] N. Etemadi, “An Elementary Proof of the Strong Law of Large Numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 55, No. 1, 1981, pp. 119-122. [3] R. G. Laha and V. K. Rohatgi, “Probability Theory,” John Wiley & Sons, Hoboken, 1979. [4] T. Birkel, “A Note on the Strong Law of Large Numbers for Positively Dependent Random Variables,” Statistics & Probability Letters, Vol. 7, No. 1, 1989, pp. 17-20. doi:10.1016/0167-7152(88)90080-6 [5] H. A. Azarnoosh, “On the Law of Large Numbers for Negatively Dependent Random Variables,” Pakistan Journal of Statistics, Vol. 19, No. 1, 2003, pp. 15-23. [6] H. R. Nili Sani, H. A. Azarnoosh and A. Bozorgnia, “The Strong Law of Large Numbers for Pairwise Negatively Dependent Random Variables,” Iranian Journal of Science & Technology, Vol. 28, No. A2, 2004, pp. 211-217. [7] T. S. Kim, H. Y. Beak and H. Y. Seo, “On Strong Laws of Large Numbers for 2-Dimensional Positively Dependent Random Variables,” Bulletin of the Korean Mathematical Society, Vol. 35, No. 4, 1998, pp. 709-718. [8] T. S. Kim, H. Y. Beak and K. H. Han, “On the Almost Sure Convergence of Weighted Sums of 2-Dimensional Arrays of Positive Dependent Random Variables,” Communications of the Korean Mathematical Society, Vol. 14, No. 4, 1999, pp. 797-804. [9] F. M′oricz, “The Kronecker Lemmas for Multiple Series and Some Applications,” Acta Mathematica Academiae Scientiarum Hungaricae, Vol. 37, No. 1-3, 1981, pp. 39- 50. doi:10.1007/BF01904871 [10] N. Ebrahimi and M. Ghosh, “Multivariate Negative Dependence,” Communications in Statistics—Theory and Methods, Vol. A10, No. 4, 1981, pp. 307-337. [11] K. L. Chung, “A Course in Probability Theory,” Academic Press, London, 2001.