Approximation of distributions of local times (Q1059931)

Denote by \(\xi\) (x,n), \(x\in Z\), \(n=0,1,2,...\), the number of visits of an aperiodic one-dimensional recurrent random walk at the point x in time n. If the random walk has finite third moment then \[ P\{\xi (x,n)=0\}=G(| x| /\sigma \sqrt{n})+O(n^{-1/6}), \] \[ P\{\xi (x,n)/\sigma^{-1}\sqrt{n}<r\}=G(| x| /\sigma \sqrt{n})+\sqrt{2/\pi}\int^{r}_{0}\exp \{-2^{-1}(| x| /\sigma \sqrt{n}+u)^ 2\}du+O(n^{-1/6}),\quad r>0. \] Here \(G(u)=\sqrt{2/\pi}\int^{u}_{0}e^{-v^ 2/2} dv\), \(u>0\).