zbMATH — the first resource for mathematics

A quenched central limit theorem for reversible random walks in a random environment on \(\mathbb{Z}\). (English) Zbl 1408.60021
Summary: The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on \(\mathbb{Z}\) without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by J. Depauw and J.-M. Derrien [C. R., Math., Acad. Sci. Paris 347, No. 7–8, 401–406 (2009; Zbl 1161.60034)]. More precisely, for a given realization \(\omega\) of the environment, we consider the Poisson equation \((P_\omega - I)g = f\), and then use the pointwise ergodic theorem in [N. Wiener, Duke Math. J. 5, 1–18 (1939; Zbl 0021.23501; JFM 65.0516.04)] 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.

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60K37 Processes in random environments
60J27 Continuous-time Markov processes on discrete state spaces
60J60 Diffusion processes
Full Text: DOI Euclid
[1] Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Prob. 36, 334-349. · Zbl 0946.60046
[2] Billingsley, P. (1995). Probability and Measure , 3rd edn. John Wiley, New York. · Zbl 0822.60002
[3] Depauw, J. and Derrien, J.-M. (2009). Variance limite d’une marche aléatoire réversible en milieu aléatoire sur \(\mathbb{Z}\). C. R. Math. Acad. Sci. Paris 347, 401-406. · Zbl 1161.60034
[4] Kawazu, K. and Kesten, H. (1984). On birth and death processes in symmetric random environment. J. Statist. Phys. 37, 561-576. · Zbl 0587.60088
[5] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40, 61-120, 238. · Zbl 0592.60054
[6] Mathieu, P. (2008). Quenched invariance principles for random walks with random conductances. J. Statist. Phys. 130, 1025-1046. · Zbl 1214.82044
[7] 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. · Zbl 0486.60076
[8] Wiener, N. (1939). The ergodic theorem. Duke Math. J. 5, 1-18. · Zbl 0021.23501
[9] Zeitouni, O. (2006). Random walks in random environments. J. Phys. A 39, R433-R464. · Zbl 1108.60085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.