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A note on the law of the iterated logarithm in Hilbert space
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  • Published: June 1989

A note on the law of the iterated logarithm in Hilbert space

  • Uwe Einmahl1 nAff2 

Probability Theory and Related Fields volume 82, pages 213–223 (1989)Cite this article

Summary

Kuelbs (1975) established a Kolmogorov-Erdös-Petrowski type integral test for lower and upper classes in the law of the iterated logarithm for sums of i.i.d. Hilbert space valued Gaussian mean zero random variables. We show that this integral test remains valid for sums of i.i.d. pregaussian mean zero random variables satisfying an additional (very mild) assumption.

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References

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

    Google Scholar 

  2. Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc.6, 170–176 (1955)

    Google Scholar 

  3. Einmahl, U.: Strong invariance principles for partial sums of independent random vectors. Ann. Probab.15, 1419–1440 (1987a)

    Google Scholar 

  4. Einmahl, U.: A useful estimate in the multidimensional invariance principle. Probab. Th. Rel. Fields76, 81–101 (1987b)

    Google Scholar 

  5. Einmahl, U.: The Darling-Erdös theorem for sums of i.i.d. random variables. Probab. Th. Rel. Fields (in press) (1989)

  6. Feller, W.: The law of the iterated logarithm for identically distributed random variables. Ann. Math.47, 631–638 (1946)

    Google Scholar 

  7. Kuelbs, J.: Sample path behavior for Brownian motion in Banach spaces. Ann. Probab.3, 247–261 (1975)

    Google Scholar 

  8. Kuelbs, J., Kurtz, T.: Berry-Esseen estimates in Hilbert space and an application to the law of the iterated logarithm. Ann. Probab.2, 387–407 (1974)

    Google Scholar 

  9. Yurinskii, V.V.: Exponential inequalities for sums of random vectors. J. Multivariate Anal.6, 473–499 (1976)

    Google Scholar 

  10. Zolotarev, V.M.: Concerning a certain probability problem. Theor. Probab. Appl.6, 201–204 (1961)

    Google Scholar 

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Author information

Author notes
  1. Uwe Einmahl

    Present address: Department of Mathematics, Indiana University, 47405, Bloomington, IN, USA

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, 48824-1027, East Lansing, MI, USA

    Uwe Einmahl

Authors
  1. Uwe Einmahl
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Einmahl, U. A note on the law of the iterated logarithm in Hilbert space. Probab. Th. Rel. Fields 82, 213–223 (1989). https://doi.org/10.1007/BF00354760

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  • Received: 23 March 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00354760

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Keywords

  • Hilbert Space
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Iterate Logarithm
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