# A general law of iterated logarithm

• Rainer Wittmann
Article

## Summary

We show that the law of iterated logarithm holds for a sequence of independent random variables (X n ) 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)

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.

## Keywords

Stochastic Process Probability Theory Mathematical Biology Classical Result Stability Result
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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