Acta Mathematica Sinica, English Series

, Volume 25, Issue 2, pp 209–222 | Cite as

Laws of the iterated logarithm for locally square integrable martingales

Article
First Online:
Received:
Revised:
Accepted:

Abstract

Three types of laws of the iterated logarithm (LIL) for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.

Keywords

locally square integrable martingales law of the iterated logarithm 

MR(2000) Subject Classification

60F15 60G44 60F17 60G42 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Lepingle, D.: Sur la comportement asymptotique des martingales locales. Lecture notes in Math., 649, 148–161 (1976)CrossRefMathSciNetGoogle Scholar
  2. [2]
    Wang, J. G.: The asymptotic behavior of locally square integrable martingales. Ann. Probab., 23, 552–585 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Xu, Y.: The law of iterated logarithm for locally square integrable martingales. Chinese J Appl. Probab. Statist., 6, 290–301 (1990)Google Scholar
  4. [4]
    Besdziek, N.: Strong approximations of semimartingales by processes with independent increments. Probab. Th. Rel. Fields 87, 489–520 (1991)MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Fisher, E.: On the law of the iterated logarithm for martingales. Ann. Probab. 20, 675–680 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Gao, F. Q.: Chung’s law of the iterated logarithm for locally square integrable martingales. Chinese Science Bulletin, 43 2156–2161 (1998)Google Scholar
  7. [7]
    Zheng, M.: The Chung law of the iterated logarithm for locally square integrable martingales. Chinese J. Appl. Probab. Stat., 14, 250–257 (1998)MATHGoogle Scholar
  8. [8]
    He, S. W., Wang, J. G., Yan, J. A.: Semimartingale Theory and Stochastic Calculus. Science Press & CRC Press Inc., Beijing & Boca Raton, 1992MATHGoogle Scholar
  9. [9]
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987MATHGoogle Scholar
  10. [10]
    Liptser, R. Sh., Shiryaev, A. N.: Theorey of Martingales, Kluwer Academic, 1989Google Scholar
  11. [11]
    Einmahl, U., Mason, D. M.: Darling-Erdös theorems for martingales. Journal of Theoretical Probability, 2, 437–460 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Einmahl, U., Mason, D. M.: Some results on the almost sure behavior of martingales. Colloquia Mathematica Societatis Janos Bolyai, 57, 185–195 (1990)MathSciNetGoogle Scholar
  13. [13]
    Hall, P., Heyde, C. C.: Martingale Limit Theory and its Applications, Academic Press, New York, 1980Google Scholar
  14. [14]
    Jain, N. C., Jogeo, K., Stout, W. F.: Upper and lower class functions for martingales and mixing processes. Ann. Probab., 3, 119–145 (1975)MATHCrossRefGoogle Scholar
  15. [15]
    Martikainen, A. I.: Chung’s law for nonidentically distributed independent summands. Theor. Probab. Appl., 31, 566–568 (1986)MathSciNetGoogle Scholar
  16. [16]
    Philipp, W., Stout, W.: Invariance principles for martingales and sums of independent random variables. Math. Z., 192, 253–264 (1986)MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Shorack, G. R., Wellner, J.: Empirical Process with Applications to Statistics, John Wiley & Sons Inc., New York, 1986Google Scholar
  18. [18]
    Csörgö, M., Révész, P.: Strong Approximations in Probability and statistics, Academic, New York, 1981MATHGoogle Scholar
  19. [19]
    Jain, N. C., Pruitt, W. E.: The other law of the iterated logarithm. Ann. Probab., 3, 1046–1049 (1975)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanP. R. China

Personalised recommendations