Journal of Theoretical Probability

, Volume 21, Issue 3, pp 704–744 | Cite as

The LIL for U-Statistics in Hilbert Spaces



We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest.


U-statistics Tail and moment estimates Law of the iterated logarithm 

Mathematics Subject Classification (2000)

60F15 60E15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamczak, R.: Moment inequalities for U-statistics. Ann. Probab. 34, 2288–2314 (2005) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Adamczak, R., Latała, R.: LIL for canonical U-statistics. Ann. Probab. (2008, to appear). Available online at
  3. 3.
    Arcones, M., Giné, E.: On the law of the iterated logarithm for canonical U-statistics and processes. Stoch. Process. Appl. 58, 217–245 (1995) CrossRefMATHGoogle Scholar
  4. 4.
    Bousquet, O., Boucheron, S., Lugosi, G., Massart, P.: Moment inequalities for functions of independent random variables. Ann. Probab. 33, 514–560 (2005) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    de la Peña, V.H., Giné, E.: Decoupling. From Dependence to Independence. Springer, New York (1999) Google Scholar
  6. 6.
    de la Peña, V.H., Montgomery-Smith, S.: Bounds for the tail probabilities of U-statistics and quadratic forms. Bull. Am. Math. Soc. 31, 223–227 (1994) CrossRefMATHGoogle Scholar
  7. 7.
    de la Peña, V.H., Montgomery-Smith, S.J.: Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23, 806–816 (1995) CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Giné, E., Latała, R., Zinn, J.: Exponential and moment inequalities for U-statistics. In: High Dimensional Probability II. Progress in Probability, vol. 47, pp. 13–38. Birkhauser, Boston (2000) Google Scholar
  9. 9.
    Giné, E., Zhang, C.-H.: On integrability in the LIL for degenerate U-statistics. J. Theor. Probab. 9, 385–412 (1996) CrossRefMATHGoogle Scholar
  10. 10.
    Giné, E., Zinn, J.: A remark on convergence in distribution of U-statistics. Ann. Probab. 22, 117–125 (1994) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Giné, E., Kwapień, S., Latała, R., Zinn, J.: The LIL for canonical U-statistics of order 2. Ann. Probab. 29, 520–557 (2001) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Gluskin, E.D., Kwapień, S.: Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Stud. Math. 114, 303–309 (1995) MATHGoogle Scholar
  13. 13.
    Goodman, V., Kuelbs, J., Zinn, J.: Some results on the LIL in Banach space with applications to weighted empirical processes. Ann. Probab. 9, 713–752 (1981) CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Houdré, C., Reynaud-Bouret, P.: Exponential inequalities, with constants, for U-statistics of order two. In: Stochastic inequalities and applications. Progress in Probability, vol. 56, pp. 55–69. Birkhauser, Basel (2003) Google Scholar
  15. 15.
    Kwapień, S., Latała, R., Oleszkiewicz, K., Zinn, J.: On the limit set in the law of the iterated logarithm for U-statistics of order two. In: High dimensional probability, III. Progress in Probability, vol. 55, pp. 111–126. Birkhauser, Basel (2003) Google Scholar
  16. 16.
    Latała, R.: On the almost sure boundedness of norms of some empirical operators. Stat. Probab. Lett. 38, 177–182 (1998) CrossRefGoogle Scholar
  17. 17.
    Latała, R.: Estimation of moments and tails of Gaussian chaoses. Ann. Probab. 34, 2061–2440 (2006) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Latała, R., Zinn, J.: Necessary and sufficient conditions for the strong law of large numbers for U-statistics. Ann. Probab. 28, 1908–1924 (2000) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Rubin, H., Vitale, R.A.: Asymptotic distribution of symmetric statistics. Ann. Stat. 8, 165–170 (1980) CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Institute of MathematicsWarsaw UniversityWarszawaPoland

Personalised recommendations