Improved results in almost sure central limit theorems for the maxima and partial sums for Gaussian sequences

详细信息    查看全文

• 作者：Qunying Wu (1)
  
  1. College of Science ; Guilin University of Technology ; Guilin ; 541004 ; P.R. China
  
• 关键词：60F15 ; standardized Gaussian sequence ; maxima ; partial sums and maxima ; almost sure central limit theorem
• 刊名：Journal of Inequalities and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：944 KB
• 参考文献：1. Brosamler, GA (1988) An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 104: pp. 561-574 CrossRef
  2. Schatte, P (1988) On strong versions of the central limit theorem. Math. Nachr. 137: pp. 249-256 CrossRef
  3. Ibragimov, IA, Lifshits, M (1998) On the convergence of generalized moments in almost sure central limit theorem. Stat. Probab. Lett. 40: pp. 343-351 CrossRef
  4. Miao, Y (2008) Central limit theorem and almost sure central limit theorem for the product of some partial sums. Proc. Indian Acad. Sci. Math. Sci. 118: pp. 289-294 CrossRef
  5. Berkes, I, Cs谩ki, E (2001) A universal result in almost sure central limit theory. Stoch. Process. Appl. 94: pp. 105-134 CrossRef
  6. H枚rmann, S (2007) Critical behavior in almost sure central limit theory. J. Theor. Probab. 20: pp. 613-636 CrossRef
  7. Wu, QY (2011) Almost sure limit theorems for stable distribution. Stat. Probab. Lett. 81: pp. 662-672 CrossRef
  8. Wu, QY (2011) An almost sure central limit theorem for the weight function sequences of NA random variables. Proc. Indian Acad. Sci. Math. Sci. 121: pp. 369-377 CrossRef
  9. Wu, QY (2012) A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law. J. Inequal. Appl. 2012: CrossRef
  10. Wu, QY, Chen, PY (2013) An improved result in almost sure central limit theorem for self-normalized products of partial sums. J. Inequal. Appl. 2013: CrossRef
  11. Fahrner, I, Stadtm眉ller, U (1998) On almost sure max-limit theorems. Stat. Probab. Lett. 37: pp. 229-236 CrossRef
  12. Nadarajah, S, Mitov, K (2002) Asymptotics of maxima of discrete random variables. Extremes 5: pp. 287-294 CrossRef
  13. Cs谩ki, E, Gonchigdanzan, K (2002) Almost sure limit theorems for the maximum of stationary Gaussian sequences. Stat. Probab. Lett. 58: pp. 195-203 CrossRef
  14. Chen, SQ, Lin, ZY (2006) Almost sure max-limits for nonstationary Gaussian sequence. Stat. Probab. Lett. 76: pp. 1175-1184 CrossRef
  15. Tan, ZQ, Peng, ZX, Nadarajah, S (2007) Almost sure convergence of sample range. Extremes 10: pp. 225-233 CrossRef
  16. Tan, ZQ, Peng, ZX (2009) Almost sure convergence for non-stationary random sequences. Stat. Probab. Lett. 79: pp. 857-863 CrossRef
  17. Peng, ZX, Liu, MM, Nadarajah, S (2008) Conditions based on conditional moments for max-stable limit laws. Extremes 11: pp. 329-337 CrossRef
  18. Peng, ZX, Wang, LL, Nadarajah, S (2009) Almost sure central limit theorem for partial sums and maxima. Math. Nachr. 282: pp. 632-636 CrossRef
  19. Peng, ZX, Weng, ZC, Nadarajah, S (2010) Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences. Extremes 13: pp. 463-480 CrossRef
  20. Zhao, SL, Peng, ZX, Wu, SL (2010) Almost sure convergence for the maximum and the sum of nonstationary Gaussian sequences. J. Inequal. Appl. 2010: CrossRef
  21. Tan, ZQ, Wang, YB (2011) Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences. J.聽Korean Stat. Soc. 40: pp. 347-355 CrossRef
  22. Chandrasekharan, K, Minakshisundaram, S (1952) Typical Means. Oxford University Press, Oxford
  23. Chuprunov, A, Fazekas, I (1999) Almost sure versions of some analogues of the invariance principle. Publ. Math. (Debr.) 54: pp. 457-471
  24. Chuprunov, A, Fazekas, I (2002) Almost sure versions of some functional limit theorems. J. Math. Sci. (N.Y.) 111: pp. 3528-3536 CrossRef
  25. Fazekas, I, Rychlik, Z (2002) Almost sure functional limit theorems. Ann. Univ. Mariae Curie-Sk艂odowska, Sect. A 56: pp. 1-18
  26. Leadbetter, MR, Lindgren, G, Rootz茅n, H (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York
  27. Seneta, E (1976) Regularly Varying Functions. Springer, Berlin
  28. Bingham, NH, Goldie, CM, Teugels, JL (1987) Regular Variation. Cambridge University Press, Cambridge CrossRef
  
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

Let \(X, X_{1}, X_{2},\ldots\) be a standardized Gaussian sequence. The universal results in almost sure central limit theorems for the maxima \(M_{n}\) and partial sums and maxima \((S_{n}/\sigma_{n}, M_{n})\) are established, respectively, where \(S_{n}=\sum_{i=1}^{n}X_{i}\) , \(\sigma^{2}_{n}=\operatorname{Var}S_{n}\) , and \(M_{n}=\max_{1\leq i\leq n}X_{i}\) .