# High Dimensional Probability VIBracketing Entropy of High Dimensional Distributions

High Dimensional Probability VI: Bracketing Entropy of High Dimensional Distributions [Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} be the class of probability distribution functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[0,\,1]^{d},\,{d}\geq{2}$$ \end{document}. The following estimate for the bracketing entropy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[L]^{p}$$\end{document} norm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\,\leq\,p\,{<} \infty$$ \end{document}, is obtained: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\rm log}{N_{[\,]}}(\varepsilon, \mathcal{F}_{d},{\parallel.\parallel}_p)=O(\varepsilon^{-1}{|\rm log\varepsilon|^{2(\rm d-1)}}).$$ \end{document} Based on this estimate, a general relation between bracketing entropy in the Lp norm and metric entropy in the L1 norm for multivariate smooth functions is established.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# High Dimensional Probability VIBracketing Entropy of High Dimensional Distributions

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

/lp/springer-journals/high-dimensional-probability-vi-bracketing-entropy-of-high-dimensional-OLu1z0jo4Z

# References (15)

Publisher
Springer Basel
ISBN
978-3-0348-0489-9
Pages
3 –17
DOI
10.1007/978-3-0348-0490-5_1
Publisher site
See Chapter on Publisher Site

### Abstract

[Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} be the class of probability distribution functions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[0,\,1]^{d},\,{d}\geq{2}$$ \end{document}. The following estimate for the bracketing entropy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_{d}$$ \end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[L]^{p}$$\end{document} norm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\,\leq\,p\,{<} \infty$$ \end{document}, is obtained: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\rm log}{N_{[\,]}}(\varepsilon, \mathcal{F}_{d},{\parallel.\parallel}_p)=O(\varepsilon^{-1}{|\rm log\varepsilon|^{2(\rm d-1)}}).$$ \end{document} Based on this estimate, a general relation between bracketing entropy in the Lp norm and metric entropy in the L1 norm for multivariate smooth functions is established.]

Published: Apr 1, 2013

Keywords: Bracketing entropy; metric entropy; high dimensional distribution