# High Dimensional Probability VIRates of Convergence in the Strong Invariance Principle for Non-adapted Sequences Application to Ergodic Automorphisms of the Torus

High Dimensional Probability VI: Rates of Convergence in the Strong Invariance Principle for... [In this paper, we give rates of convergence in the strong invariance principle for non-adapted sequences satisfying projective criteria. The results apply to the iterates of ergodic automorphisms T of the d-dimensional torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{T}^d$$ \end{document}, even in the non hyperbolic case. In this context, we give a large class of unbounded function f from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{T}^d$$ \end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}, for which the partial sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f\;\circ\;T\;+\;f\;\circ\;T^{2}\;+\;.\;.\;.\;+\;f\;\circ\;T^{n}$$ \end{document} satisfies a strong invariance principle with an explicit rate of convergence.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# High Dimensional Probability VIRates of Convergence in the Strong Invariance Principle for Non-adapted Sequences Application to Ergodic Automorphisms of the Torus

Part of the Progress in Probability Book Series (volume 66)
Editors: Houdré, Christian; Mason, David M.; Rosiński, Jan; Wellner, Jon A.
26 pages

/lp/springer-journals/high-dimensional-probability-vi-rates-of-convergence-in-the-strong-wUa5HVHZUK

# References (29)

Publisher
Springer Basel
ISBN
978-3-0348-0489-9
Pages
113 –138
DOI
10.1007/978-3-0348-0490-5_9
Publisher site
See Chapter on Publisher Site

### Abstract

[In this paper, we give rates of convergence in the strong invariance principle for non-adapted sequences satisfying projective criteria. The results apply to the iterates of ergodic automorphisms T of the d-dimensional torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{T}^d$$ \end{document}, even in the non hyperbolic case. In this context, we give a large class of unbounded function f from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{T}^d$$ \end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}, for which the partial sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f\;\circ\;T\;+\;f\;\circ\;T^{2}\;+\;.\;.\;.\;+\;f\;\circ\;T^{n}$$ \end{document} satisfies a strong invariance principle with an explicit rate of convergence.]

Published: Apr 1, 2013

Keywords: Almost sure invariance principle; strong approximations; nonadapted sequences; ergodic automorphisms of the torus