Skip to main content
SpringerLink
Log in

LIL behavior for B-valued strong mixing random variables

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Given a sequence of mixing random variables {X, X n ; n ≥ 1} taking values in a separable Banach space B, and S n denoting the partial sum, a general law of the iterated logarithm is established, that is, we have with probability one, lim supn→∞ |S n |/c n = α 0 < ∞ for a regular normalizing sequence {c n }, where α 0 is a precise value.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acosta A De, Kuelbs J. Some results on the cluster set C({S n/a n}) and the LIL. Ann Probab, 1983, 11: 102–122

    Article  MATH  MathSciNet  Google Scholar 

  2. Berkes I, Philipp W. Approximation theorems for independent and weakly dependent random vectors. Ann Probab, 1979, 7: 29–54

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkes I, Philipp W. Limits theorems for mixing sequences without rate assumptions. Ann Probab, 1998, 26: 805–831

    Article  MATH  MathSciNet  Google Scholar 

  4. Einmahl U. Toward a general law of the iterated logarithm in Banach space. Ann Probab, 1993, 21: 2012–2045

    Article  MATH  MathSciNet  Google Scholar 

  5. Einmahl U, Kuelbs J. Cluster sets for a generalized law of the iterated logarithm in Banach spaces. Ann Probab, 2001, 29: 1451–1475

    Article  MATH  MathSciNet  Google Scholar 

  6. Einmahl U, Li D L. Some results on two-sided LIL behavior. Ann Probab, 2005, 33: 1601–1624

    Article  MATH  MathSciNet  Google Scholar 

  7. Einmahl U, Li D L. Characterization of the LIL behavior in Banach space. Trans Amer Math Soc, 2008, 360: 6677–6693

    Article  MATH  MathSciNet  Google Scholar 

  8. Feller W. An extension of the law of the iterated logarithm to variables without variance. J Math Mech, 1968, 18: 343–355

    MATH  MathSciNet  Google Scholar 

  9. Kesten H. Sums of independent random variables-without moment conditions. Ann Math Statist, 1972, 43: 701–732

    Article  MATH  MathSciNet  Google Scholar 

  10. Klass M. Toward a universal law of the iterated logarithm I. Z Wahrsch Verw Gebiete, 1976, 36: 165–178

    Article  MATH  MathSciNet  Google Scholar 

  11. Klass M. Toward a universal law of the iterated logarithm II. Z Wahrsch Verw Gebiete, 1977, 39: 151–165

    Article  MATH  MathSciNet  Google Scholar 

  12. Klass M. The finite mean LIL bounds are sharp. Ann Probab, 1984, 12: 907–911

    Article  MATH  MathSciNet  Google Scholar 

  13. Kuelbs J. When is the cluster set of S n/a n empty? Ann Probab, 1981, 9: 377–394

    Article  MATH  MathSciNet  Google Scholar 

  14. Kuelbs J. The LIL when X is in the domain of attraction of a Gaussian law. Ann Probab, 1985, 13: 825–859

    Article  MATH  MathSciNet  Google Scholar 

  15. Kuelbs J, Zinn J. Some results on LIL behavior. Ann Probab, 1983, 11: 506–557

    Article  MATH  MathSciNet  Google Scholar 

  16. Lin Z Y, Lu C R. Limit Theory for Mixing Random Variables. Dordrecht/Beijing: Kluwer Academic Publishers/Science Press, 1997

    Google Scholar 

  17. Liu W D, Fu K A, Zhang L X. A LIL for independent non-identically distributed random variables in Banach space and its applications. Sci China Ser A, 2008, 51: 219–232

    Article  MATH  MathSciNet  Google Scholar 

  18. Sharipov O Sh. Probabilistic characterization of Banach spaces by weakly dependent random elements (in Russian). Uzbek Mat Zh, 1998, 2: 86–91

    MathSciNet  Google Scholar 

  19. Sharipov O Sh. The law of the iterated logarithm for weakly dependent Hilbert space valued random variables. Sib Math J, 2003, 44: 1111–1126

    Article  MathSciNet  Google Scholar 

  20. Sharipov O Sh. The bounded law of the iterated logarithm for weakly dependent random variables with values in a type 2 Banach space. Theory Probab Appl, 2005, 49: 444–457

    Article  MathSciNet  Google Scholar 

  21. Zhang L X. Rosenthal type inequalities for B-valued strong mixing random fields and their applications. Sci China Ser A, 1997, 41: 736–745

    Article  Google Scholar 

  22. Zhang L X, Wen J W. Strong limit theorems for B-valued random fields (in Chinese). Chinese Ann Math, 2001, 22A: 205–216

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China

    KeAng Fu

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    LiXin Zhang

Authors
  1. KeAng Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. LiXin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to KeAng Fu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fu, K., Zhang, L. LIL behavior for B-valued strong mixing random variables. Sci. China Math. 54, 785–792 (2011). https://doi.org/10.1007/s11425-011-4196-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-011-4196-7

Keywords

MSC(2000)

Search

Navigation