DOI QR코드

DOI QR Code

LIMSUP RESULTS AND LIL FOR PARTIAL SUMS OF RANDOM SEQUENCES

  • Han, Chang-Ho (Department of Mathematics and RINS, Gyeongsang National University) ;
  • Moon, Hee-Jin (Department of Mathematics and RINS, Gyeongsang National University) ;
  • Choi, Yong-Kab (Department of Mathematics and RINS, Gyeongsang National University)
  • Received : 2013.08.09
  • Accepted : 2013.12.03
  • Published : 2014.01.31

Abstract

In this paper we establish limsup results and a generalized uniform law of the iterated logarithm (LIL) for the increments of partial sums of strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) random sequences.

Keywords

References

  1. T. Birkel, A Functional Central Limit Theorem for Positively Dependent Random Variables, J. Multivariate Anal. 44 (1993), no. 2, 314-320. https://doi.org/10.1006/jmva.1993.1018
  2. Y. K. Choi and M. Csorgo, Path properties of $l^{p}$-valued Gaussian random fields, Sci. China Ser. A: Math. 50 (2007), 1501-1520. https://doi.org/10.1007/s11425-007-0084-6
  3. Y. K. Choi and M. Csorgo, Limsup results and LIL for partial sum processes of a Gaussian random field, Acta Math. Sinica. 24 (2008), no. 9, 1497-1506. https://doi.org/10.1007/s10114-008-6205-5
  4. Y. K. Choi, K. S. Hwang, T. S. Kim, Z. Y. Lin and W. S. Wang, Asymptotic behaviors for partial sum processes of a Gaussian sequence, Acta Math. Hungar. 103 (2004), 43-54. https://doi.org/10.1023/B:AMHU.0000028235.82111.cf
  5. M. Csorgo, Z. Y. Lin, and Q. M. Shao, Path properties for $l^{{\infty}}$-valued Gaussian processes, Proc. Amer. Math. Soc. 121 (1994), 225-236.
  6. M. Csorgo and P. Revesz, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981.
  7. M. Csorgo and P. Revesz, How big are the increments of a Wiener process? Ann. Probab. 7 (1979), 731-737. https://doi.org/10.1214/aop/1176994994
  8. M. Csorgo and P. Revesz , How small are the increments of a Wiener process? Stochastic Process Appl. 8 (1979), 119-129.
  9. M. Csorgo and J. Steinebach, Improved Erdos-Renyi and strong approximation laws for increments of partial sums, Ann. Probab. 9 (1981), no. 6, 988-996. https://doi.org/10.1214/aop/1176994269
  10. P. Deheuvels and J. Steinebach, Exact convergence rates in strong approximation laws for large increments of partial sums, Probab. Theory Related Fields 76 (1987), 369-393. https://doi.org/10.1007/BF01297492
  11. P. Erdos and A. Renyi, On a new law of large numbers, J. Analyse Math. 13 (1970), 103-111.
  12. J. Esary, F. Proschan and D.Walkup, Association of random variables with applications, Ann. Math. Statist. 38 (1967), no. 4, 1466-1474. https://doi.org/10.1214/aoms/1177698701
  13. K. Joag-Dev, Independence via uncorrelatedness under certain dependence structures, Ann. Probab. 11 (1983), no. 4, 1037-1041. https://doi.org/10.1214/aop/1176993452
  14. 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
  15. H. Lanzinger and U. Stadtmuller, Maxima of increments of partial sums for certain subexponential distributions, Stochastic Process. Appl. 86 (2000), 307-322. https://doi.org/10.1016/S0304-4149(99)00100-3
  16. E. L. Lehmann, Some concepts od dependence, Ann. Math. Statist. 37 (1966), no. 5, 1137-1153. https://doi.org/10.1214/aoms/1177699260
  17. Y. X. Li and J. F. Wang, The law of the iterated logarithm for positively dependent random variables, J. Math. Anal. Appl. 339 (2008), no. 1, 259-265. https://doi.org/10.1016/j.jmaa.2007.06.044
  18. Z. Y. Lin, On Csorgo-Revesz's increments of sums of non-i.i.d. random variables, Scientia Sinica (Series A) 30 (1987), 921-931.
  19. Z. Y. Lin, On the increments of partial sums of ${\phi}$-mixing sequence, Teor. Veroyatnost. i Primenen. 36 (1991), 326-336; translation in Theory Probab. Appl. 36 (1992), 316-328.
  20. Z. Y. Lin, S. H. Lee, K. S. Hwang and Y. K. Choi, Some limit theorems on the increments of $l^{p}$-valued multi-parameter Gaussian processes, Acta Math. Sinica, English Ser. 20 (2004), no. 6, 1019-1028. https://doi.org/10.1007/s10114-004-0414-3
  21. Z. Y. Lin and C. R. Lu, Strong Limit Theorems, New York, 1975.
  22. Z. Y. Lin, C. R. Lu and L. X. Zhang, Path Properties of Gaussian Processes, Zhejiang University Press, 2001.
  23. C. M. Newman, Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 74 (1980), no. 2, 119-128. https://doi.org/10.1007/BF01197754
  24. C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability (Tong Y. L., Ed., Institute of Mathematical Statistics, Hayward, CA) (1984), 127-140.
  25. V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, New York, 1975.
  26. J. F. Wang and L. X. Zhang, A Berry-Esseen theorem for weakly negatively dependent random variables and its applications, Acta Math. Hungar. 110 (2006), no, 4, 293-308. https://doi.org/10.1007/s10474-006-0024-x
  27. Y. Yang and Y. B. Wang, The asymptotical normality of the renewal process generated by strictly stationary LPQD sequences, Chinese J. Appl. Probab. Statist. 24 (2008), no. 1, 37-42.
  28. L. X. Zhang, Central limit theorems for asymptotically negatively associated random fields, Acta Math. Sinica, Engl. Ser. 16 (2000), no. 4, 691-710. https://doi.org/10.1007/s101140000084