  In this note, we prove a sharp $L^p$-rate of convergence of the number of
upcrossings to the local time of the Brownian motion. In particular, it
provides novel $p$-variation estimates ($2 < p < \infty$) for the number of
upcrossings of the Brownian motion. Our result complements the fundamental work
of Koshnevisan \cite{kho} who obtains an almost sure exact rate of convergence
in the sup norm.
