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The purpose of this note is to prove a central limit theorem for the$L^3$-modulus of continuity of the Brownian local time using techniques ofstochastic analysis. The main ingredients of the proof are an asymptoticversion of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulusof the Brownian local time.

Central limit theorem for the modulus of continuity of the Brownian local time in $L^3(\mathbb{R})$