대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제54권3호
  • /
  • Pages.839-874
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  • 2017
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON SOME APPLICATIONS OF THE ARCHIMEDEAN COPULAS IN THE PROOFS OF THE ALMOST SURE CENTRAL LIMIT THEOREMS FOR CERTAIN ORDER STATISTICS

  • Dudzinski, Marcin (Faculty of Applied Informatics and Mathematics Warsaw University of Life Sciences - SGGW w Warszawie) ;
  • Furmanczyk, Konrad (Faculty of Applied Informatics and Mathematics Warsaw University of Life Sciences - SGGW w Warszawie)
  • 투고 : 2016.04.16
  • 발행 : 2017.05.31
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초록

Our goal is to establish and prove the almost sure central limit theorems for some order statistics $\{M_n^{(k)}\}$, $k=1,2,{\ldots}$, formed by stochastic processes ($X_1,X_2,{\ldots},X_n$), $n{\in}N$, the distributions of which are defined by certain Archimedean copulas. Some properties of generators of such the copulas are intensively used in our proofs. The first class of theorems stated and proved in the paper concerns sequences of ordinary maxima $\{M_n\}$, the second class of the presented results and proofs applies for sequences of the second largest maxima $\{M_n^{(2)}\}$ and the third (and the last) part of our investigations is devoted to the proofs of the almost sure central limit theorems for the k-th largest maxima $\{M_n^{(k)}\}$ in general. The assumptions imposed in the first two of the mentioned groups of claims significantly differ from the conditions used in the last - the most general - case.

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참고문헌

  1. I. Berkes and E. Csaki, A universal result in almost sure central limit theory, Stochastic Process. Appl. 94 (2001), no. 1, 105-134. https://doi.org/10.1016/S0304-4149(01)00078-3
  2. G. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 3, 561-574. https://doi.org/10.1017/S0305004100065750
  3. S. Chen and Z. Lin, Almost sure max-limits for nonstationary Gaussian sequence, Statist. Probab. Lett. 76 (2006), no. 11, 1175-1184. https://doi.org/10.1016/j.spl.2005.12.018
  4. S. Cheng, L. Peng, and Y. Qi, Almost sure convergence in extreme value theory, Math. Nachr. 190 (1998), 43-50. https://doi.org/10.1002/mana.19981900104
  5. E. Csaki and K. Gonchigdanzan, Almost sure limit theorems for the maximum of sta- tionary Gaussian sequences, Statist. Probab. Lett. 58 (2002), 195-203. https://doi.org/10.1016/S0167-7152(02)00128-1
  6. M. Dudzinski, A note on the almost sure central limit theorem for some dependent random variables, Statist. Probab. Lett. 61 (2003), 31-40. https://doi.org/10.1016/S0167-7152(02)00291-2
  7. M. Dudzinski, The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences, Statist. Probab. Lett. 78 (2008), 347-357. https://doi.org/10.1016/j.spl.2007.07.007
  8. M. Dudzinski and P. Gorka, The almost sure central limit theorems for the maxima of sums under some new weak dependence assumptions, Acta Math. Sin. (Engl. Ser.) 29 (2013), 429-448. https://doi.org/10.1007/s10114-013-1388-9
  9. I. Fazekas and Z. Rychlik, Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Sk lodowska, Sect. A 56 (2002), 1-18.
  10. E. W. Frees and E. A. Valdez, Understanding relationships using copulas, N. Am. Actuar. J. 2 (1998), no. 1, 1-25. https://doi.org/10.1080/10920277.1998.10595667
  11. K. Gonchigdanzan and G. Rempa la, A note on the almost sure limit theorem for the product of partial sums, Appl. Math. Lett. 19 (2006), no. 2, 191-196. https://doi.org/10.1016/j.aml.2005.06.002
  12. S. Hormann, A note on the almost sure convergence of central order statistics, Probab. Math. Statist. 25 (2005), 317-329.
  13. C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152-164. https://doi.org/10.1007/BF01832852
  14. M. Lacey and W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), no. 3, 201-205. https://doi.org/10.1016/0167-7152(90)90056-D
  15. A. W. Marshall and I. Olkin, Families of multivariate distributions, J. Amer. Statist. Assoc. 83 (1988), no. 403, 834-841. https://doi.org/10.1080/01621459.1988.10478671
  16. P. Matu la, On almost sure central limit theorem for associated random variables, Probab. Math. Statist. 18 (1998), no. 2, 411-416.
  17. A. J. McNeil and J. Neslehova, Available on the following website, http://www- 1.ms.ut.ee/tartu07/presentations/mcneil.pdf.
  18. A. J. McNeil and J. Neslehova, Multivariate Archimedean copulas, d-monotone functions and ${\ell}_1$-norm sym-metric distributions, Ann. Statist. 37 (2009), no. 5B, 3059-3097. https://doi.org/10.1214/07-AOS556
  19. J. Mielniczuk, Some remarks on the almost sure central limit theorem for dependent sequences, In: Limit theorems in Probability and Statistics II (I. Berkes, E. Csaki, M. Csorgo, eds.), Bolyai Institute Publications, Budapest, pp. 391-403, 2002.
  20. M. Peligrad and Q. Shao, A note on the almost sure central limit theorem for weakly dependent random variables, Statist. Probab. Lett. 22 (1995), no. 2, 131-136. https://doi.org/10.1016/0167-7152(94)00059-H
  21. L. Peng and Y. Qi, Almost sure convergence of the distributional limit theorem for order statistics, Probab. Math. Statist. 23 (2003), no. 2, 217-228.
  22. Z. Peng, J. Li, and S. Nadarajah, Almost sure convergence of extreme order statistics, Electron. J. Statist. 3 (2009), 546-556. https://doi.org/10.1214/08-EJS303
  23. P. Schatte, On the central limit theorem with almost sure convergence, Probab. Math. Statist. 11 (1991), 237-246.
  24. A. Sklar, Fonctions de repartition r n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231.
  25. A. Sklar, Random variables, distribution functions and Copulas - a personal look back- ward and forward, In: Ruschendorf, L., Schweizer, B., Taylor, M.D. (Eds.), Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Hayward, CA, pp. 1-14, 1996.
  26. U. Stadtmuller, Almost sure versions of distributional limit theorems for certain order statistics, Statist. Probab. Lett. 58 (2002), no. 4, 413-426. https://doi.org/10.1016/S0167-7152(02)00156-6
  27. Z. Tan, Almost sure central limit theorem for exceedance point processes of stationary sequences, Braz. J. Probab. Statist. 29 (2015), no. 3, 717-731. https://doi.org/10.1214/14-BJPS242
  28. M. V. Wuthrich, Extreme value theory and Archimedean copulas, Scand. Actuar. J. 104 (2004), no. 3, 211-228.
  29. S. Zhao, Z. Peng, and S. Wu, Almost sure convergence for the maximum and the sum of nonstationary Guassian sequences, J. Inequal. Appl. 2010 (2010), Art. ID 856495, 14 pp.