scholarly journals Statistical stability and linear response for random hyperbolic dynamics

2021 ◽  
pp. 1-30
Author(s):  
DAVOR DRAGIČEVIĆ ◽  
JULIEN SEDRO
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Linear Response ◽  
Ad Hoc ◽  
Transfer Operator ◽  
Statistical Stability ◽  
Hyperbolic Dynamics ◽  
Quenched Central Limit Theorem ◽  
Random Products

Abstract We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

scholarly journals Quenched central limit theorem in a corner growth setting

2018 ◽  
Vol 23 ◽  
Author(s):  
H. Christian Gromoll ◽  
Mark W. Meckes ◽  
Leonid Petrov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
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 Large deviations and central limit theorems for sequential and random systems of intermittent maps

2020 ◽  
pp. 1-28
Author(s):  
MATTHEW NICOL ◽  
FELIPE PEREZ PEREIRA ◽  
ANDREW TÖRÖK
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Systems ◽  
Central Limit Theorems ◽  
Random Dynamical Systems ◽  
Stat Phys ◽  
Quenched Central Limit Theorem

Abstract We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

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.

scholarly journals Quenched Normal Approximation for Random Sequences of Transformations

2019 ◽  
Vol 178 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Olli Hella ◽  
Mikko Stenlund
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convergence Rate ◽  
Central Limit ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Central Limit Theorems ◽  
Random Sequences ◽  
Quenched Central Limit Theorem ◽  
Product Measures

AbstractWe study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate, also in the multivariate case, assuming fiberwise centering. For the most part we work with non-stationary randomness and non-invariant, non-product measures. Independently, we believe our work sheds light on the mechanisms that make quenched central limit theorems work, by dissecting the problem into three separate parts.

scholarly journals A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees

2018 ◽  
Vol 55 (2) ◽  
pp. 610-626 ◽  
Author(s):  
Adam Bowditch
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Upper Bound ◽  
Functional Central Limit Theorem ◽  
Biased Random Walk ◽  
Quenched Central Limit Theorem ◽  
Functional Central Limit

AbstractIn this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Sign in / Sign up

Export Citation Format

Share Document