Law of the iterated logarithm

Let $X_1,X_2,\dots$ be independent and identically distributed real random variables with $\mathbb{E}{X}_1=0$ and $\mathrm{Var}(X_1)={\sigma }^2\in (0,\infty )$. Write $S_n={\sum }_{j=1}^n{X}_j$. The iid law of the iterated logarithm statesFinite variance is sufficient in this iid theorem; the lower conclusion follows by applying the upper theorem to $-X_j$. The iterated logarithm is the slowly growing factor $\log \log n$.^[ref]

For a simple symmetric random walk, ${\sigma }^2=1$. Its scaled partial sums exceed every fixed multiple of $\sqrt{n}$ along some times almost surely, even though the central limit theorem gives a tight normal limit for $S_n/\sqrt{n}$ at each growing deterministic time. The iterated-logarithm law concerns the extremes encountered along one infinite realization, rather than only the distribution at one time.

It also implies $S_n/n\to 0$, since $\sqrt{\log \log n/n}\to 0$, and therefore refines the strong law of large numbers. For the sample mean, the almost-sure envelope is $\sigma \sqrt{2\log \log n/n}$. The upper law of the iterated logarithm and Brownian upper law of the iterated logarithm articles isolate one-sided parts of the argument; proving a summable upper-tail bound alone does not establish the matching lower excursions.

For standard Brownian motion, the Brownian reflection principle bounds the Probability that its maximum up to $a^n$ exceeds $(1+\epsilon )\sqrt{2a^n\log \log (a^n)}$ by $2(n\log a{)}^{-(1+\epsilon {)}^2}$. This is summable for every $a>1$ and $\epsilon >0$. The Borel-Cantelli first lemma controls these geometric times, monotonicity of the normalizing function controls the intervening times, and countable choices $a\downarrow 1$, $\epsilon \downarrow 0$ give the bound. Equality requires a separate lower-bound proof.

For a simple symmetric random walk, geometric blocking and the Borel-Cantelli first lemma yield the upper estimate ${\mathrm{limsup}}_n{S}_n/\sqrt{2n\log \log n}\le 1$ almost surely. Equality requires a separate lower-bound argument.

For a simple symmetric random walk, almost surely