Stability and instability of local time of random walk in random environment
type: Item
instance of: scholarly article
title: Stability and instability of local time of random walk in random environment
zbMATH Open document ID: 0634.60062
DOI: 10.1016/0304-4149(87)90197-9
author: Lajos Horváth
author: Pál Révész
author: Miklós Csörgo
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 4032739
zbMATH Keywords: local time
zbMATH Keywords: laws of iterated logarithm
zbMATH Keywords: random walk in random environment
zbMATH Keywords: weak limit theorem
reviewed by: Oleg K. Zakusilo
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90197-9
OpenAlex ID: W1973160755
cites work: On the zero \(\sum_1^n\pm 1\)
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description: scientific article; zbMATH DE number 4032739