A CLT for the $L^2$ norm of increments of local times of Lévy processes as time goes to infinity

Abstract

Let $X=\{X_t,t\in R_{+}\}$ be a symmetric Lévy process with local time $\{L_t^x;(x,t)\in R^1\times R_{+}^1\}$. When the Lévy exponent $ψ(\la)$ is regularly varying at zero with index $1<\beta \le 2$, 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{ψ^{-1}(1/t)}}\label{r5.0tweaksabs} && \stackrel{\mathcal{L}}{\Longrightarrow}(8c_{ψ,1 })^{1/2}(\int_{-\infty}^{\finfty} (L_{β,1}^{x})^{2} dx)^{1/2} η\end{eqnarray*} as $t\rar\infty$, where $L_{\bb,1}=\{L^{x}_{β, 1} ; x \in R^{1} \}$ 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 $\psi$.

References in corpus (2)

• $L^p$ moduli of continuity of Gaussian processes and local times of symmetric Lévy processes
• A CLT for the L^{2} modulus of continuity of Brownian local time