Journal of Theoretical Probability

, Volume 21, Issue 3, pp 704–744

The LIL for U-Statistics in Hilbert Spaces

Article

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

60F15 60E15 

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References

  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 http://arxiv.org/abs/math/0604262
  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

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