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We give a stochastic calculus proof of the Central Limit Theorem \[ {\int(L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}}\stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} \eta\]as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $\eta$ is an independentnormal random variable with mean zero and variance one.

A stochastic calculus proof of the CLT for the L^{2} modulus of continuity of local time