An invariance principle for the law of the iterated logarithm

Abstract

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 {Mathematical expression} 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 {Mathematical expression} and {Mathematical expression} As applications we obtain, e. g., {Mathematical expression} for any a ≧ 1, and {Mathematical expression} Where vn is the frequency of the events {Mathematical expression} among the first n integers i (0 ≦ c≦ 1). © 1964 Springer-Verlag.