scholarly journals Almost sure invariance principle for dynamical systems by spectral methods

2010 ◽  
Vol 38 (4) ◽  
pp. 1639-1671 ◽  
Author(s):  
Sébastien Gouëzel
Keyword(s):  
Dynamical Systems ◽  
Invariance Principle ◽  
Spectral Methods ◽  
Almost Sure Invariance Principle

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
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