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

DOI10.1214/08-AOP410zbMath1176.37006arXivmath/0606535MaRDI QIDQ1019086

Matthew Nicol, Ian Melbourne

Publication date: 27 May 2009

Published in: The Annals of Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0606535

zbMATH Keywords

Brownian motionmixing propertynonuniform hyperbolicityYoung towersalmost sure invariance principleLorentz gasessums of weakly independent variables

Mathematics Subject Classification ID

Dynamical systems and their relations with probability theory and stochastic processes (37A50) Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) (37D20) Functional limit theorems; invariance principles (60F17) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) (37D25)

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Cites Work

• Unnamed Item
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• Unnamed Item
• Almost sure invariance principle for nonuniformly hyperbolic systems
• An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables
• Almost sure invariance principles for partial sums of mixing B-valued random variables
• Statistical properties of Lorentz gas with periodic configuration of scatterers
• Ergodic properties of invariant measures for piecewise monotonic transformations
• Upper and lower functions for martingales and mixing processes
• Distribution function inequalities for martingales
• Approximation theorems for independent and weakly dependent random vectors
• Statistical properties of dynamical systems with some hyperbolicity
• Recurrence times and rates of mixing
• Statistical limit theorems for suspension flows
• Almost sure invariance principle for dynamical systems with stretched exponential mixing rates
• Central limit theorems and invariance principles for time-one maps of hyperbolic flows
• Limit laws and recurrence for the planar Lorentz process with infinite horizon
• Advanced statistical properties of dispersing billiards
• Méthode de martingales et flot géodésique sur une surface de courbure constante négative
• Statistical properties of two-dimensional hyperbolic billiards
• Approximation by Brownian motion for Gibbs measures and flows under a function
• Some Limit Theorems for Random Functions. I
• A maximal 𝕃_{𝕡}-inequality for stationary sequences and its applications
• Rapid decay of correlations for nonuniformly hyperbolic flows
• Almost sure invariance principles for partial sums of weakly dependent random variables
• Singular hyperbolic systems
• Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions
• Limit theorems for partially hyperbolic systems
• LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS
• An almost sure Invariance Principle for Hilbert Space Valued Martingales
• A new method to prove strassen type laws of invariance principle. II
• Billiards with polynomial mixing rates
• An invariance principle for the law of the iterated logarithm
• Local Convergence of Martingales and the Law of Large Numbers
• Central limit theorems and invariance principles for Lorenz attractors
• Differentiable dynamical systems
• Skorohod embedding of multivariate RV's, and the sample DF