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

We give a stochastic calculus proof of the Central Limit Theorem $$\frac{\int (L_t^{x+h}-L_t^x{)}^{2}dx-4ht}{h^{3\mathrm{/}2}}c(\int (L_t^x{)}^{2}dx{)}^{1\mathrm{/}2}\eta$$ as $h\to 0$ for Brownian local time $L_t^x$. Here $\eta$ is an independent normal random variable with mean zero and variance one.