scholarly journals A vector-valued almost sure invariance principle for hyperbolic dynamical systems

2009 ◽  
Vol 37 (2) ◽  
pp. 478-505 ◽  
Author(s):  
Ian Melbourne ◽  
Matthew Nicol
Keyword(s):  
Dynamical Systems ◽  
Invariance Principle ◽  
Hyperbolic Dynamical Systems ◽  
Almost Sure Invariance Principle ◽  
Vector Valued

scholarly journals Rates in almost sure invariance principle for slowly mixing dynamical systems

2019 ◽  
Vol 40 (9) ◽  
pp. 2317-2348 ◽  
Author(s):  
C. CUNY ◽  
J. DEDECKER ◽  
A. KOREPANOV ◽  
F. MERLEVÈDE
Keyword(s):  
Markov Chain ◽  
Dynamical Systems ◽  
Invariance Principle ◽  
Slowly Varying Function ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Deterministic Dynamical Systems ◽  
Almost Sure Invariance Principle ◽  
The One ◽  
Dependent Processes

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.

scholarly journals Almost sure invariance principle for sequential and non-stationary dynamical systems

2017 ◽  
Vol 369 (8) ◽  
pp. 5293-5316 ◽  
Author(s):  
Nicolai Haydn ◽  
Matthew Nicol ◽  
Andrew Török ◽  
Sandro Vaienti
Keyword(s):  
Dynamical Systems ◽  
Invariance Principle ◽  
Almost Sure Invariance Principle

scholarly journals A coupling approach to random circle maps expanding on the average

2014 ◽  
Vol 14 (04) ◽  
pp. 1450008 ◽  
Author(s):  
Mikko Stenlund ◽  
Henri Sulku
Keyword(s):  
Invariance Principle ◽  
Exponential Convergence ◽  
Memory Loss ◽  
Coupling Scheme ◽  
Uniform Bounds ◽  
Circle Maps ◽  
Exponential Mixing ◽  
Coupling Approach ◽  
Almost Sure Invariance Principle ◽  
Vector Valued

We study random circle maps that are expanding on the average. Uniform bounds on neither expansion nor distortion are required. We construct a coupling scheme, which leads to exponential convergence of measures (memory loss) and exponential mixing. Leveraging from the structure of the associated correlation estimates, we prove an almost sure invariance principle for vector-valued observables. The motivation for our paper is to explore these methods in a non-uniform random setting.

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