Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE EXPONENTIAL INEQUALITY FOR NEGATIVE DEPENDENT SEQUENCE

  • Published : 2007.04.30
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Abstract

We show an exponential inequality for negatively associated and strictly stationary random variables replacing an uniform boundedness assumption by the existence of Laplace transforms. To obtain this result we use a truncation technique together with a block decomposition of the sums. We also identify a convergence rate for the strong law of large number.

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References

  1. K. Azuma, Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), 357-367 https://doi.org/10.2748/tmj/1178243286
  2. M. Carbon, Une nouvelle inegalite de grandes deviations Applications, Publ. IRMA Lille 32 XI, 1993
  3. L. Devroye, Exponential inequalities in nonparametric estimation, In : Roussas, G. (Ed.) Nonparametric Functional Estimation and Related Topics. Kluwer Academic Publishers, Dordrecht, pp. 31-44, (1991)
  4. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), 286-295 https://doi.org/10.1214/aos/1176346079
  5. E. Lesigne and D. Volny, Large deviations for martingales, Stochastic. Process. Appl. 96 (2001), 143-159 https://doi.org/10.1016/S0304-4149(01)00112-0
  6. H. Y. Liang, Complete convergence for weighted sums of negatively associated ranldom variables, Statist. Prob. Lett. 48 (2000), 317-325 https://doi.org/10.1016/S0167-7152(00)00002-X
  7. P. Matula, A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Prob. Lett. 15 (1992), 209-213 https://doi.org/10.1016/0167-7152(92)90191-7
  8. C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, In: Tong, Y. L.(Ed.), Inequalities in Statistics and Probability. IMS Lectures Notes-Monograph Series 5 (1984), 127-140, Hayward CA
  9. G. G. Roussas, Exponential probability inequalities with some applications, In:Ferguson, T. S., Shapely, L. S., MacQueen, J. B.(Ed.),Statistics, Probability and Game Theory. IMS Lecture Notes-Monograph Series, 30 (1996), 303-309, Hayward, CA
  10. Q. M. Shao, A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theor. Probab. 13 (2000), 343-356 https://doi.org/10.1023/A:1007849609234
  11. Q. M. Shao and C. Su, The law of the iterated logarithm for negatively associated random variables, Stoch. Process. 83 (1999), 139-148 https://doi.org/10.1016/S0304-4149(99)00026-5

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