대한수학회지 (Journal of the Korean Mathematical Society)

  • 제45권6호
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  • Pages.1601-1611
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  • 2008
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS

  • Hwang, Kyo-Shin (THE RESEARCH INSTITUTE OF NATURAL SCIENCE GYEONGSANG NATIONAL UNIVERSITY)
  • 발행 : 2008.11.01
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초록

Let X, $X_1$, $X_2$, ... be i.i.d. random variables with zero means, variance one, and set $S_n\;=\;{\sum}^n_{i=1}\;X_i$, $n\;{\geq}\;1$. Gut and $Sp{\check{a}}taru$ [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P($|S_n|\;{\geq}\;{\varepsilon}{\sqrt{2n\;{\log}\;{\log}\;n}}$) is replaced by E{$|S_n|/{\sqrt{n}}-{\varepsilon}{\sqrt{2\;{\log}\;{\log}\;n}$}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.

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참고문헌

  1. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27. Cambridge University Press, Cambridge, 1987
  2. Y. S. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sin. (N.S.) 16 (1988), no. 3, 177-201
  3. A. Gut and A. Spataru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000), no. 4, 1870-1883 https://doi.org/10.1214/aop/1019160511
  4. C. C. Heyde, A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975), no. 1, 173-175 https://doi.org/10.2307/3212424
  5. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA 33 (1947), no. 2, 25-31c https://doi.org/10.1073/pnas.33.2.25
  6. M. L. Katz, The probability in the tail of a distribution, Ann. Math. Statist. 34 (1963), no. 1, 312-318 https://doi.org/10.1214/aoms/1177704268
  7. D. Li, B.-E. Nguyen, and A. Rosalsky, A supplement to precise asymptotics in the law of the iterated logarithm, J. Math. Anal. Appl. 302 (2005), no. 1, 84-96 https://doi.org/10.1016/j.jmaa.2004.08.009
  8. T.-X. Pang, Z.-Y. Lin, Y. Jiang, and K.-S. Hwang, Precise rates in the law of the logarithm for the moment convergence of i.i.d. random variables, J. Koran Math. Soc. 45 (2008), no. 4, 993-1005 https://doi.org/10.4134/JKMS.2008.45.4.993
  9. A. Spataru, Precise asymptotics in Spitzer's law of large numbers, J. Theoret. Probab. 12 (1999), no. 3, 811-819 https://doi.org/10.1023/A:1021636117551
  10. Q. Wang and B. Y. Jing, An exponential nonuniform Berry-Esseen bound for selfnormalized sums, Ann. Probab. 27 (1999), no. 4, 2068-2088 https://doi.org/10.1214/aop/1022677562