# An invariance principle for the law of the iterated logarithm

• V. Strassen
Article

## Summary

Let Sn 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 ηn on «0, 1» obtained by linearly interpolating (2 n log log n)−1/2Si at i/n. Then we prove (theorem 3) that with probability one the set of limit points of the sequence (ηn)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 vn is the frequency of the events
$$S_i > c(2i\log \log i)^{1/2}$$
among the first n integers i (0 ≦ c≦ 1).

## Keywords

Continuous Function Stochastic Process Probability Theory Mathematical Biology Limit Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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