An invariance principle for the law of the iterated logarithm

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).