A CLT for the L² norm of increments of local times of L\'evy processes as time goes to infinity

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 $\psi$.