= Gaussian random walk spends zero density of time in bounded sets
{title2=$n^{-1}\sum_{j\leq n}\mathbf1_B(S_j)\to0$}
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 $\mathbb P(S_j\in B)\leq |B|/\sqrt{2\pi j}$. Consequently $\mathbb E\sum_j\mathbf1_B(S_j)/j<\infty$, 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.
Back to article page
