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An invariance principle for the law of the iterated logarithm
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  • Published: September 1964

An invariance principle for the law of the iterated logarithm

  • V. Strassen1,2 nAff3 

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete volume 3, pages 211–226 (1964)Cite this article

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Summary

Let S n be the sum of the first n of a sequence of independent identically distributed r. v. s. having mean 0 and variance 1. One version of the law of the iterated logarithm asserts that with probability one the set of limit points of the sequence

$$((2n{\text{ log }}\log n)^{ - 1/2} S_n )_{n \geqq 3} $$

coincides with «-1, 1» = {x:x real and ¦x¦≦ 1} (see Hartman-Wintner [6]). Now consider the continuous function η n on «0, 1» obtained by linearly interpolating (2 n log log n)−1/2 S i at i/n. Then we prove (theorem 3) that with probability one the set of limit points of the sequence (η n)n≧3 with respect to the uniform topology coincides with the set of absolutely continuous functions x on «0, 1» such that

$$x(0) = 0$$

and

$$\int {\dot x^2 dt \leqq 1} $$

As applications we obtain, e. g.,

$$Pr\left\{ {\mathop {{\text{lim sup}}}\limits_{n \to \infty } n^{ - 1 - (a/2)} (2{\text{ log log }}n)^{ - (a/2)} \sum\limits_{i = 1}^n {|S_i |a} = \frac{{2(a + 2)^{(a/2) - 1} }}{{\left( {\int\limits_0^1 {\frac{{dt}}{{\sqrt {1 - t^a } }}} } \right)^a a^{a/2} }}} \right\} = 1$$

for any a ≧ 1, and

$$Pr\left\{ {\mathop {{\text{lim sup}}}\limits_{n \to \infty } v_n = 1 - {\text{exp}}\left\{ { - 4\left( {\frac{1}{{c^2 }} - 1} \right)} \right\}} \right\} = 1$$

Where v n is the frequency of the events

$$S_i > c(2i\log \log i)^{1/2} $$

among the first n integers i (0 ≦ c≦ 1).

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References

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  2. Diedonné, I.: Foundations of Modern Analysis, Pure and Applied Mathematics. New York-London: 1960.

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  5. Feller, W.: The general form of the so-called law of the iterated logarithm. Trans. Amer. Math. Soc., 54, 373–402 (1943).

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  6. Hartman, P., and A. Wintner: On the law of the iterated logarithm. Amer. J. Math., 63, 169–176 (1941).

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Author information

Author notes
  1. V. Strassen

    Present address: Inst. f. Math. Stat. der UniversitÄt, Bürgerstr. 32., 34 Göttingen

Authors and Affiliations

  1. University of California, Berkeley

    V. Strassen

  2. UniversitÄt Göttingen, Deutschland

    V. Strassen

Authors
  1. V. Strassen
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Additional information

I would like to thank Professor M. Loève for directing my attention to the book of Skorokhod and Professor D.Freedman for pointing out an error in the original manuscript.

This paper was prepared with the partial support of the Office of Naval Research, Contract Nonr-222-43. This paper in whole or in part may be reproduced for any purpose of the U.S. Government.

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Cite this article

Strassen, V. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw Gebiete 3, 211–226 (1964). https://doi.org/10.1007/BF00534910

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  • Received: 20 July 1964

  • Issue Date: September 1964

  • DOI: https://doi.org/10.1007/BF00534910

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Keywords

  • Continuous Function
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Limit Point
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