Gaussian random walk spends zero density of time in bounded sets ($n^{-1}{\sum }_{j\le n}{1}_B(S_j)\to 0$)

For independent $N(0,1)$ increments, a random walk has $S_j\sim N(0,j)$. If $B$ is a bounded measurable set, its occupation time probability obeys $\mathbb{P}(S_j\in B)\le |B|/\sqrt{2\pi j}$. Consequently $\mathbb{E}{\sum }_j{1}_B(S_j)/j<\infty$, and Tonelli's theorem makes this series finite almost surely. Kronecker lemma then gives the displayed zero occupation density. This also shows why its empirical occupation probabilities cannot converge to a probability measure simultaneously on all bounded intervals and the whole real line: continuity from below would contradict the preserved total mass one.