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