Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

Antoniou, Marios; Melbourne, Ian

doi:10.1007/s00220-019-03334-6

Cited by 14 publications

(28 citation statements)

References 46 publications

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“…As examples, we mention the martingale CLT/WIP [13] (see Appendix A) and an ASIP for reverse martingale differences [21] discussed at the end of this section. A third application [9] is to the estimate of convergence rates in the WIP. To control the Birkhoff sums of φ, a martingale-coboundary decomposition for φ is of great utility; this is the topic of the current section.…”

Section: Secondary Martingale-coboundary Decomposition and The Asipmentioning

confidence: 99%

Martingale–coboundary decomposition for families of dynamical systems

Korepanov

Kosloff

Melbourne

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractWe prove statistical limit laws for sequences of Birkhoff sums of the typewhere T n is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T n is replaced by a fixed transformation T , and which is particularly effective in the case when T n varies with n.In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

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Section: Secondary Martingale-coboundary Decomposition and The Asipmentioning

confidence: 99%

Martingale–coboundary decomposition for families of dynamical systems

Korepanov

Kosloff

Melbourne

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…Homogenization (convergence of fast-slow systems to a stochastic differential equation) when the fast dynamics is given by f follows from [9,20,32]. Convergence rates in the WIP and homogenization are obtained in [4].…”

Section: Some Statistical Propertiesmentioning

confidence: 99%

Sharp Statistical Properties for a Family of Multidimensional NonMarkovian Nonconformal Intermittent Maps

Eslami¹,

Melbourne²,

Vaienti³

2019

Preprint

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Intermittent maps of Pomeau-Manneville type are well-studied in onedimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theorem and local limit theorem, Berry-Esseen estimates, large deviation estimates, and convergence to stable laws and Lévy processes.

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“…Homogenization (convergence of fast-slow systems to a stochastic differential equation) when the fast dynamics is one of these maps f : M → M follows from [19,23,35]. Convergence rates in the WIP and homogenization are obtained in [4].…”

Section: Upper Bounds and Limit Lawsmentioning

confidence: 99%