A Stochastic Calculus Proof of the CLT for the L2 Modulus of Continuity of Local Time

  • Jay RosenEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


We give a stochastic calculus proof of the Central Limit Theorem
$${ \int \nolimits \nolimits {({L}_{t}^{x+h} - {L}_{t}^{x})}^{2}\,dx - 4ht \over {h}^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c{\left (\int \nolimits \nolimits {({L}_{t}^{x})}^{2}\,dx\right )}^{1/2}\,\,\eta $$
as h → 0 for Brownian local time L t x . Here η is an independent normal random variable with mean zero and variance one.
Central limit theorem Moduli of continuity Local time Brownian motion 


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This research was supported, in part, by grants from the National Science Foundation and PSC-CUNY.


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsCollege of Staten Island, CUNYStaten IslandUSA

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