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We give a stochastic calculus proof of the Central Limit Theorem [( ò\nolimits \nolimits (Ltx+h - Ltx)2 dx - 4ht)/(h3/2)] Þ Lc(ò\nolimits \nolimits (Ltx)2 dx)1/2  h{ \int \nolimits \nolimits {({L}_{t}^{x+h} - {L}_{t}^{x})}^{2}\,dx - 4ht \over {h}^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c{\left (\int \nolimits \nolimits {({L}_{t}^{x})}^{2}\,dx\right )}^{1/2}\,\,\eta as h → 0 for Brownian local time L t x . Here η is an independent normal 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