Law of the iterated logarithm

For independent identically distributed centered variables of variance one under standard moment assumptions, the law of the iterated logarithm describes the almost-sure envelope of their partial sums on the scale $\sqrt{2n\log \log n}$.

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