On the inverse local time process of a plane random walk (Q5933712)

Let \(S_{n}\) \((n=0,1,2,\dots)\) be a simple symmetric random walk on the lattice \(\mathbb Z^2\). Introduce the following notation: \[ \rho_{0}=0,\quad \rho_{k}=\min\{n: n>\rho_{k-1},\;S_{n}=0\},\;k\geq 1,\quad T_{n}=(\rho_{n+1}-\rho_{n})/\rho_{n}. \] \textit{E. Csaki, P. Révész} and \textit{Z. Shi} [Long excursions of a random walk (to appear)] studied the properties of the sequence \(\{T_{n}\}\). The main result claimes that \(T_{n}\) is either very small or very large, roughly speaking. In the present paper some further properties of \(T_{n}\) are given. For example, the next theorem tells us that \(T_{n}\), with a big probability, is very small: \[ \mathbb P\{T_{n}<\exp(-n/z)\}=1-\exp(-\pi z)(1+O(n^{-1/7})) \] uniformly for \(0<z<n^{3/7}\). Another Theorem claims that \(T_{n}\) occasionally is very large. Also, the limit distribution of \(T_{n}\) is evaluated when \(T_{n}\) is large and the strong limit behaviour of \(T_{n}\) is described. A strong approximation of the sequence \(\{\rho_{n}\}\) by a relatively simple process is given. The well-known quantile-transformation method is applied.