Occupation approximations for Brownian local time ($L_t^0={\lim }_{n\to \infty }\frac{n}{2}{\int }_0^t{1}_{\{|B_s|<1/n\}}ds$)

For Brownian motion started at zero, let $f_n$ equal $|x|$ outside $(-1/n,1/n)$ and $n{x}^2/2+1/(2n)$ inside. Its derivative is the clipped sign function $g_n(x)=\max \{-1,\min \{1,nx\}\}$. The Itô integral $M_t^{(n)}={\int }_0^t{g}_n(B_s)d{B}_s$ converges uniformly on bounded time intervals in $L^2$ to a continuous martingale $M$, by the Itô isometry and Doob L2 maximal inequality: the squared maximal error is at most $8\sqrt{2T/\pi }/n$. The Itô formula gives $\frac{n}{2}{\int }_0^t{1}_{\{|B_s|<1/n\}}ds=f_n(B_t)-f_n(0)-M_t^{(n)}$. Along $n=k^2$, the maximal errors are summable, so First Borel-Cantelli lemma gives uniform almost-sure convergence. The ratio-one interpolation for shrinking-window occupation integrals extends it to every integer $n$. The limit is $|B_t|-M_t=L_t^0$, the local time of a semimartingale in the Tanaka formula convention. The unnormalized factor $n$ gives twice this limit.

For any measurable real path $x$, put $A_n(t)=n{\int }_0^t{1}_{\{|x(s)|<1/n\}}ds$. If $n_k\le n\le n_{k+1}$, nested windows give $(n/n_{k+1})A_{n_{k+1}}(t)\le A_n(t)\le (n/n_k)A_{n_k}(t)$. Consequently, when $n_{k+1}/n_k\to 1$ and the subsequence $A_{n_k}$ converges uniformly on bounded time intervals, the full sequence does also, to the same limit. This deterministic interpolation turns summable-subsequence First Borel-Cantelli lemma estimates into almost-sure convergence without incorrectly claiming that convergence in $L^2$ alone implies full-sequence almost-sure convergence.