A Central Limit Theorem for Time-Dependent Dynamical Systems

@article{Nndori2012ACL,
  title={A Central Limit Theorem for Time-Dependent Dynamical Systems},
  author={P{\'e}ter N{\'a}ndori and D. Sz{\'a}sz and T. Varj{\'u}},
  journal={Journal of Statistical Physics},
  year={2012},
  volume={146},
  pages={1213-1220}
}
The work by Ott et al. (Math. Res. Lett. 16:463–475, 2009) established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case… Expand
Central limit theorems with a rate of convergence for time-dependent intermittent maps
We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-likeExpand
A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems
We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advanceExpand
Central Limit Theorems with a Rate of Convergence for Dynamical Systems
Central limit theorems are some of the most classical theorems in the theory of probability. They have also been actively studied in the field of dynamical systems. In the first article of thisExpand
Limit theorems for random expanding or hyperbolic dynamical systems and vector-valued observables.
The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establishExpand
Sunklodas’ Approach to Normal Approximation for Time-Dependent Dynamical Systems
We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariateExpand
Robustness of ergodic properties of non-autonomous piecewise expanding maps
Recently, there has been an increasing interest in non-autonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviourExpand
Statistical properties of non-stationary dynamical systems with intermittency
The problems addressed in this thesis revolve around two types of non-stationary dynamical systems: sequential compositions of interval maps with a neutral fixed point (Pomeau-Manneville maps) andExpand
Intermittent quasistatic dynamical systems: weak convergence of fluctuations
Abstract This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over timeExpand
Intermittent quasistatic dynamical systems: weak convergence of fluctuations
This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due toExpand
Quasistatic dynamical systems
We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes inExpand
...
1
2
3
...

References

SHOWING 1-10 OF 13 REFERENCES
Limit theorems for sequential expanding dynamical systems on [0,1]
We consider the asymptotic behaviour of a sequence (theta(n)), theta(n) = tau(n) o tau(n - 1) . . . o tau(1), where (tau(n))(n >= 1) are non-singular transformations on a probability space. AfterExpand
Non-stationary compositions of Anosov diffeomorphisms
Motivated by non-equilibrium phenomena in nature, we study dynamical systems whose time-evolution is determined by non-stationary compositions of chaotic maps. The constituent maps are topologicallyExpand
RANDOM PROCESSES GENERATED BY A HYPERBOLIC SEQUENCE OF MAPPINGS. II
For a sequence of smooth mappings of a Riemannian manifold, which is a nonstationary analogue of a hyperbolic dynamical system, a compatible sequence of measures carrying one into another under theExpand
Quenched CLT for random toral automorphism
We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for theExpand
Brownian Brownian Motion-I
A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work the authors study a 2D version of this model, where theExpand
Central Limit Theorem for Deterministic Systems
A unified approach to obtaining the central limit theorem for hyperbolic dynamical systems is presented. It builds on previous results for one dimensional maps but it applies to the multidimensionalExpand
Higher cohomology for abelian groups of toral automorphisms II: the partially hyperbolic case, and corrigendum
In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for ${\mathbb Z}^k$-actions by hyperbolic automorphisms of a torus, to theExpand
A Martingale Proof of Dobrushin's Theorem for Non-Homogeneous Markov Chains
In 1956, Dobrushin proved an important central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given throughExpand
Central limit theorem for products of toral automorphisms
Let $(\tau_n)$ be a sequence of toral automorphisms $\tau_n : x \rightarrow A_n x \hbox{mod}\ZZ^d$ with $A_n \in {\cal A}$, where ${\cal A}$ is a finite set of matrices in $SL(d, \mathbb{Z})$. UnderExpand
Higher cohomology for Abelian groups of toral automorphisms
We give a complete description of smooth untwisted cohomology with coefficients in ℝ l for ℤ k -actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ k − 1 the nth cohomology trivializes, i.e.Expand
...
1
2
...