# Entropy conditions for subsequences of random variables with applications to empirical processes

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2008-02-01

AUTHORS ABSTRACT

.We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under the assumption that the entropy function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}$\end{document} is not too large, many strong limit theorems will continue to hold uniformly over all sequences in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}$\end{document}. We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly over all sequences in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}$\end{document}. We prove a similar result for sequences (nkω) mod 1 where the sequence (nk) of real numbers satisfies a Hadamard gap condition. More... »

PAGES

183-204

### Journal

TITLE

Monatshefte für Mathematik

ISSUE

3

VOLUME

153

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00605-007-0519-8

DOI

http://dx.doi.org/10.1007/s00605-007-0519-8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1025807658

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