A CLT for the $L^2$ moduli of continuity of local times of Levy processes

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 infinity with index $1<\beta \le 2$ and satisfies some additional regularity conditions && \sqrt{hψ^{2}(1/h)} \lc \int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E(\int (L^{x+h}_{1}- L^{x}_{1})^{2} dx)\rc\nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow} (8c_{β,1})^{1/2} η(\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h\rar 0$, where $\eta$ is a normal random variable with mean zero and variance one that is independent of $L_t^x$, and $c_{\beta ,1}$ is a known constant.

References in corpus (1)

• $L^p$ moduli of continuity of Gaussian processes and local times of symmetric Lévy processes