A CLT for the L^{2} modulus of continuity of Brownian local time

Chen, Xia; Li, Wenbo; Marcus, Michael B.; Rosen, Jay

A CLT for the L^{2} modulus of continuity of Brownian local time

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Description

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.