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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.

MSC:
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
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References:
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