Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T18:38:02.208Z Has data issue: false hasContentIssue false
  • English
  • Français

A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Hoang-Chuong Lam
Hoang-Chuong Lam*
Affiliation:
Can Tho University and Ben Gurion University
*
∗ Postal address: Department of Mathematics, Can Tho University, Can Tho City, Vietnam. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

Keywords

Quenched central limit theoremreversible random walk in random environment

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J60: Diffusion processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1051 - 1064
Copyright
© Applied Probability Trust 

References

Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Prob. 36, 334–349.CrossRefGoogle Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Depauw, J. and Derrien, J.-M. (2009). Variance limite d'une marche aléatoire réversible en milieu aléatoire sur {\BBZ}. C. R. Math. Acad. Sci. Paris 347, 401–406.CrossRefGoogle Scholar
Kawazu, K. and Kesten, H. (1984). On birth and death processes in symmetric random environment. J. Statist. Phys. 37, 561–576.CrossRefGoogle Scholar
Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40, 61–120, 238.Google Scholar
Mathieu, P. (2008). Quenched invariance principles for random walks with random conductances. J. Statist. Phys. 130, 1025–1046.CrossRefGoogle Scholar
Papanicolaou, G. C. and Varadhan, S. R. S. (1982). Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao, North-Holland, Amsterdam, pp. 547–552.Google Scholar
Wiener, N. (1939). The ergodic theorem. Duke Math. J. 5, 1–18.CrossRefGoogle Scholar
Zeitouni, O. (2006). Random walks in random environments. J. Phys. A 39, R433–R464.CrossRefGoogle Scholar