# Applications of the law of the iterated logarithm

The law of the iterated logarithm says that if $X_n$ is a sequence of iid random variables with zero expectation and unit variance, then the partial sums sequence $S_n = \sum_{i = 1}^n X_i$ satisfies almost surely that $\limsup_{n \rightarrow \infty} \frac{S_n}{\sqrt{2 n \log{\log\ n}}} = 1$.

What are the applications of this result? Why is it considered important or even useful?

I looked at the wikipedia article. It doesn't explain to me in detail where is this result applied. What major results are built from it or what major areas of applications are.

What I am looking for is something like a list of major applications of that theorem. Like how is it used to proved other stuff.

• according to wikipedia, "In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk." – Gregory Grant Dec 26 '15 at 8:14
• Did you read the "Discussion" part of the wikipedia article? It gives some detail about its theoretical importance. – Jimmy R. Dec 26 '15 at 8:53
• I've seen a few papers use it to get bounds on the liminf of a stochastic differential equation – Brenton Dec 27 '15 at 5:40