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

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

2009-09-06

arxiv.org/abs/0909.1084v1

Mathematics

Probability

Scientific paper

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 $\psi$.

Affiliated with

Also associated with

No associations

Rating

A CLT for the $L^{2}$ norm of increments of local times of Lévy processes as time goes to infinity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A CLT for the $L^{2}$ norm of increments of local times of Lévy processes as time goes to infinity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A CLT for the $L^{2}$ norm of increments of local times of Lévy processes as time goes to infinity will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-723705