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On the rate of convergence in Strassen's law of the iterated logarithm
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  • Published: April 1987

On the rate of convergence in Strassen's law of the iterated logarithm

  • Karl Grill1 

Probability Theory and Related Fields volume 74, pages 583–589 (1987)Cite this article

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  • 8 Citations

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Summary

Let W(t) be a standard Wiener process and let L be the compact class figuring in Strassen's law of the iterated logarithm. We investigate the rate of convergence to zero of the variable

$$\mathop {inf}\limits_{f \in \mathfrak{L}} {\text{ }}\mathop {{\text{sup}}}\limits_{{\text{0}} \leqq x \leqq 1} {\text{ |}}W(xT)(2T log log T)^{ - \frac{1}{2}} - f(x)|.$$

It is shown that as T→∞, (log log T)-α belongs to the upper class of this variable if α<2/3, and to the lower class if α>2/3.

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References

  1. Bolthausen, E.: On the speed of convergence in Strassen's law of the iterated logarithm. Ann. Probab. 6, 668–672 (1978)

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  2. Feller, W.: An introduction to probability theory and its applications 2. New York: Wiley 1966

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  3. Strassen, V.: An invariance principle for the law of iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb. 3, 211–216 (1964)

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Authors and Affiliations

  1. Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, Wiedner Hauptstrasse 8, A-1040, Wien, Austria

    Karl Grill

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  1. Karl Grill
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Cite this article

Grill, K. On the rate of convergence in Strassen's law of the iterated logarithm. Probab. Th. Rel. Fields 74, 583–589 (1987). https://doi.org/10.1007/BF00363517

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  • Received: 15 March 1986

  • Revised: 24 October 1986

  • Accepted: 24 October 1986

  • Issue Date: April 1987

  • DOI: https://doi.org/10.1007/BF00363517

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Wiener Process
  • Lower Class
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