  Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'{e}vy process with local time
$\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'{e}vy exponent
$\psi(\la)$ is regularly varying at zero with index $1<\beta\leq 2$, and
satisfies some additional regularity conditions, \begin{eqnarray*} &&
{\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx-
E(\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\over
t\sqrt{\psi^{-1}(1/t)}}\label{r5.0tweaksabs} &&
\stackrel{\mathcal{L}}{\Longrightarrow}(8c_{\psi,1
})^{1/2}(\int_{-\infty}^{\finfty} (L_{\beta,1}^{x})^{2} dx)^{1/2} \eta
\end{eqnarray*} as $t\rar\infty$, where $L_{\bb,1}=\{L^{x}_{\beta, 1} ; x \in
R^{1} \}$ denotes the local time, at time 1, of a symmetric stable process with
index $\beta$, $\eta$ is a normal random variable with mean zero and variance
one that is independent of $L_{\beta,1}$, and $c_{\psi,1}$ is a known constant
that depends on $\psi$.
