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The law of the iterated logarithm for empirical processes on Vapnik-Cervonenkis classes

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  • Alexander, Kenneth S.
  • Talagrand, Michel

Abstract

Necessary and sufficient conditions are established for the bounded and compact laws of the iterated logarithm for empirical processes indexed by classes of functions which have the Vapnik-Cervonenkis property. Results of Ledoux and Talagrand (Ann. Probab. 16 1242-1264, and in press) reduce the problem fromone of almostsure behavior to one of in-probability behavior. The special case of weighted empirical processes indexed by sets is considered.

Suggested Citation

  • Alexander, Kenneth S. & Talagrand, Michel, 1989. "The law of the iterated logarithm for empirical processes on Vapnik-Cervonenkis classes," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 155-166, July.
    • Handle: RePEc:eee:jmvana:v:30:y:1989:i:1:p:155-166
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    Cited by:

    1. Kawaguchi, Kohei, 2017. "Testing rationality without restricting heterogeneity," Journal of Econometrics, Elsevier, vol. 197(1), pages 153-171.
    2. Sun, Liuquan & Zhu, Lixing, 2000. "A semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 217-227, July.
    3. Arcones, Miguel A. & Giné, Evarist, 1995. "On the law of the iterated logarithm for canonical U-statistics and processes," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 217-245, August.
    4. Sun, Liuquan & Zhou, Yong, 1998. "Sequential confidence bands for densities under truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 40(1), pages 31-41, September.

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