## 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

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

and

As applications we obtain, e. g.,

for any *a* ≧ 1, and

Where *v*
_{n} is the frequency of the events

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

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

### Keywords

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