ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY SPECTRAL METHODS

By S. Gouëzel. 2009

125 citations recorded by the source corpus.

We prove the almost sure invariance principle for stationary ℝ d -valued random processes (with very precise dimension-independent error terms), solely under a strong assumption concerning the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains using strong or weak spectral perturbation arguments.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ee9b246ad9852d60e3acaa460f6edb923eb7c162

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Citations in the corpus

Smooth approximation of stochastic differential equations (2014, background): “For large classes of uniformly and nonuniformly hyperbolic flows [11, 20, 33, 35], it can be shown that Wn satisfies the WIP.”
Annealed and quenched limit theorems for random expanding dynamical systems (2013, methodology): “We apply Gouëzel’s spectral method to prove an annealed almost sure invariance principle for vector valued observables : this is a strong reinforcement of the central limit theorem which has many consequences, such as the law of the iterated logarithm, the functional central limit theorem, and the ”
Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples (2011, background): “Using chaining arguments, convexity and the Doob’s maximal inequality, as above, we infer that for any p > 1, ‖ max 0≤m≤2r−1 |E(S2r+v − S2r+v−m|F2r−m)|‖p (15)”
Almost sure invariance principle for sequential and non-stationary dynamical systems (2014, methodology): “Gouëzel [16] used spectral methods to give error rates in the ASIP for a wide class of dynamical systems, and his formulation does not require the assumption of a Young Tower.”
A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems (2017, unknown): “This spectral approach has beenwidely used to prove limit theorems for deterministic dynamics, including large deviation principles [28,44], central limit theorems [6,11,28,45], Berry-Esseen theorems [23,26], local central limit theorems [23,28,45], and vectorvalued almost-sure invariance principles”