= Occupation approximations for Brownian local time
{title2=$L_t^0=\lim_{n\to\infty}\frac n2\int_0^t\mathbf1_{\{|B_s|<1/n\}}ds$}
For <Brownian motion> started at zero, let $f_n$ equal $|x|$ outside $(-1/n,1/n)$ and $nx^2/2+1/(2n)$ inside. Its derivative is the clipped sign function $g_n(x)=\max\{-1,\min\{1,nx\}\}$. The <Itô integral> $M_t^{(n)}=\int_0^t g_n(B_s)dB_s$ 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 $\frac n2\int_0^t\mathbf1_{\{|B_s|<1/n\}}ds=f_n(B_t)-f_n(0)-M_t^{(n)}$. 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.
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