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.
 A: Since I am a statistics guy, here are some facts relating to the application of law of iterated algorithms as far as my experience.General speaking people use it to evaluate probability bound. If you know a bit about the concept "confidence interval" and "power of the statistical test", you could have very amazing results by applying law of iterated algorithms, such as the confidence interval sequences with coverage probability 1 for all sample sizes, and the famous power 1 test (which means that the type II error is zero) based on stopping rule.This is the perfect statistical test people want, however often difficult to obtain in traditional approach by fixing sample sizes and so on.For the above two points, please refer to the Sect.3 and Sect.4 of the paper by Herbert Robbins (https://projecteuclid.org/euclid.aoms/1177696786) for which is a classic paper. Sect.1 and Sect.2 is just showing how to use the theorem to deduce the favorable results.
Another famous application of the theorem is related to Bahadur representation of quantiles of distribution, referring to the paper (http://arxiv.org/pdf/math/0508313.pdf). Donot be worried about details, key is to observe the remainder term of the asymptotic expansion to see how it relates to law of iterated algorithms.
I would say it is rather theoretical and hope you enjoy it and help you a bit.:)
