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The Law of the Iterated Logarithm for Empirical Processes

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Probability in Banach Spaces 6

Part of the book series: Progress in Probability ((PRPR,volume 20))

Abstract

Let (A,A,P) be a probability space and F ⊂ L2(A,A,P). Let X i , i ≥ 1, be a sequence of i.i.d. random variables with distribution P and let P n = n−1(δX 1 + ? + δX n ) be the n’th empirical measure for P. Using the methods employed to describe functional Donsker classes, we characterize when the normalized empirical process

$${{v}_{n}}(f):={{n}^{1/2}}\int{f(d{{P}_{n}}-dP)},f\in F$$

satisfies the compact and bounded law of the iterated logarithm (LIL) uniformly over F. Sufficient conditions implying the bounded LIL are obtained. In particular, we obtain two new metric entropy integral conditions implying the bounded LIL. Moreover, the integral condition is essentially the best possible.

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Authors and Affiliations

  1. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    J. E. Yukich

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  1. J. E. Yukich
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Editors and Affiliations

  1. Matematisk Institut, Odense Universitet, DK-5230, Odense M, Denmark

    U. Haagerup  & N. J. Nielsen  & 

  2. Matematisk Institut, Aarhus Universitet, DK-8000, Aarhus C, Denmark

    J. Hoffmann-Jørgensen

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© 1990 Birkhäuser Boston

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Yukich, J.E. (1990). The Law of the Iterated Logarithm for Empirical Processes. In: Haagerup, U., Hoffmann-Jørgensen, J., Nielsen, N.J. (eds) Probability in Banach Spaces 6. Progress in Probability, vol 20. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6781-9_15

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  • DOI: https://doi.org/10.1007/978-1-4684-6781-9_15

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4684-6783-3

  • Online ISBN: 978-1-4684-6781-9

  • eBook Packages: Springer Book Archive

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