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 $$\sum\limits_{n = 1}^\infty {(s_n^2 2\log _2 s_n^2 )^{ - \frac{p}{2}} E(\left| {X_n } \right|^p )} < \infty {\text{ for a 2 < }}p \leqq 3,$$ (i) $$\mathop {\lim }\limits_{n \to \infty } s_n = \infty$$ (ii) $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{{s_{n + 1} }}{{s_n }} < \infty ,{\text{ where }}s_n : = \left( {\sum\limits_{i = 1}^n {E(X_i^2 )} } \right)^{\frac{1}{2}}$$ (iii)