scholarly journals Quenched central limit theorem for random walks in doubly stochastic random environment

2018 ◽  
Vol 46 (6) ◽  
pp. 3558-3577 ◽  
Author(s):  
Bálint Tóth
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Doubly Stochastic ◽  
Quenched Central Limit Theorem

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

2014 ◽  
Vol 51 (04) ◽  
pp. 1051-1064
Author(s):  
Hoang-Chuong Lam
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Integrability Condition ◽  
The Central Limit Theorem ◽  
The Poisson Equation ◽  
Convergence Of Moments ◽  
Quenched Central Limit Theorem

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 (P ω - 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.

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

2014 ◽  
Vol 51 (4) ◽  
pp. 1051-1064 ◽  
Author(s):  
Hoang-Chuong Lam
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Integrability Condition ◽  
The Central Limit Theorem ◽  
The Poisson Equation ◽  
Convergence Of Moments ◽  
Quenched Central Limit Theorem

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.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (01) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.

scholarly journals Quenched local central limit theorem for random walks in a time-dependent balanced random environment

2021 ◽  
Author(s):  
Jean-Dominique Deuschel ◽  
Xiaoqin Guo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Transition Probabilities ◽  
Time Dependent ◽  
Upper And Lower Bounds ◽  
Negative Moment ◽  
Local Central Limit Theorem

AbstractWe prove a quenched local central limit theorem for continuous-time random walks in $${\mathbb {Z}}^d, d\ge 2$$ Z d , d ≥ 2 , in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.

The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment

2004 ◽  
Vol 41 (1) ◽  
pp. 83-92 ◽  
Author(s):  
Jean Bérard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Random Environment ◽  
Space Time ◽  
Nearest Neighbour ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Almost All

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.

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