Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 1 / b / Solution

The law of the iterated logarithm for a simple symmetric random walk states that almost surelySince $\sqrt{2\log \log n}$ is unbounded, $|S_n|/\sqrt{n}$ exceeds every fixed $c>0$ at some finite time. Hencealmost surely.