Central limit theorems for the $L^{2}$ norm of increments of local times of Lévy processes

Abstract

Let $X=\{X_{t},t\in R_{+}\}$   be a symmetric Lévy process with  local time   $\{L^{ x }_{ t}\,;\,(x,t)\in R^{ 1}\times  R^{  1}_{ +}\}$. When the Lévy exponent $\psi(\lambda)$  is regularly varying at zero with   index $1<\beta\leq 2$, and satisfies some additional regularity conditions,  $$ \lim_{t\to\infty}{ \int_{-\infty}^{\infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2}\,dx- E\left(\int_{-\infty}^{\infty} ( L^{ x+1}_{t}- L^{ x}_{ t})^{ 2}\,dx\right)\over t\sqrt{\psi^{-1}(1/t)}}$$ is equal in law to $$(8c_{\psi,1 })^{1/2}\left(\int_{-\infty}^{\infty} \left(L_{\beta,1}^{x}\right)^{2}\,dx\right)^{1/2}\,\eta$$ where    $L_{\beta,1}=\{L^{ x }_{\beta, 1}\,;\, x \in R^{ 1} \}$ 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 $\psi$.
When the Lévy exponent $\psi(\lambda)$  is regularly varying at infinity with   index $1<\beta\leq 2$ and satisfies some additional regularity conditions $$\lim_{h\to 0}\sqrt{h\psi^{2}(1/h)} \left\{ \int_{-\infty}^{\infty} ( L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2}\,dx- E\left( \int_{-\infty}^{\infty} (  L^{ x+h}_{1}- L^{ x}_{ 1})^{ 2}\,dx\right)\right\}$$ is equal in law to $$(8c_{\beta,1})^{1/2}\,\,\eta\,\, \left( \int_{-\infty}^{\infty}  (L_{1}^{x})^{2}\,dx\right)^{1/2}$$ where $\eta$ is a normal random variable with mean zero and variance one that is independent of   $\{L^{ x }_{ 1},x\in R^{1}\}$, and $c_{\beta,1}$ is a known constant.