A general law of iterated logarithm

Abstract

We show that the law of iterated logarithm holds for a sequence of independent random variables (Xn) provided {Mathematical expression} {Mathematical expression} {Mathematical expression} The classical result of Hartman and Wintner is then a simple corollary. Another law of iterated logarithm due to Petrov is also a corollary and can even be improved. Unlike other treatments our approach is not based on the Berry-Esseen theorem. Instead we use a simple estimate of Butzer and Hahn. In this way the paper is quite elementary. As a further application of our method we prove stability results for sequences of independent random variables. © 1985 Springer-Verlag.