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The purpose of this note is to prove a central limit theorem for the$L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR},using techniques of stochastic analysis. The main ingredients of the proof arean asymptotic version of Knight's theorem and the Clark-Ocone formula for the$L^2$-modulus of the Brownian local time.

Stochastic integral representation of the $L^{2}$ modulus of Brownian local time and a central limit theorem