Stability and instability of local time of random walk in random environment
Let \(\{S_ n\}\) be a random walk in random environment \(\chi\), i.e. \[ P_{\chi}\{S_{n+1}=i+1| S_ n=i\}=X_ i,\quad P_{\chi}\{S_{n+1}=i-1| S_ n=i\}=1-X_ i, \] where \(\{X_ i\}\) is a sequence of i.i.d. random variables with \(P\{X_ i\in (0,1)\}=1\). Put \(\xi (x,n)=\#\{k:\) \(0\leq k\leq n\), \(S_ k=x\}\), \(\nu (k,n)=\xi (k,n)/\xi (0,n)\). The authors give conditions for existence of a random function \(m_ k\) (depending only on the environment \(\chi)\) such that \[ (1)\quad P_{\chi}\{\lim_{n\to \infty}\exp ((\log n)(\log \log n)^{- 1+\epsilon})| \nu (k,n)\quad -m_ k| =0\}=1\quad a.s. \] Some results concerning nonuniformity in (1) and a weak limit theorem for \(\nu (k_ n,n)\) are also given.
scientific article; zbMATH DE number 4032739