English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 $1<\beta\leq 2$, andsatisfies 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)\overt\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 $L_{\bb,1}=\{L^{x}_{\beta, 1} ; x \inR^{1} \}$ denotes the local time, at time 1, of a symmetric stable process withindex $\beta$, $\eta$ is a normal random variable with mean zero and varianceone that is independent of $L_{\beta,1}$, and $c_{\psi,1}$ is a known constantthat depends on $\psi$.

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