Statistics Papers

Document Type

Journal Article

Date of this Version

1978

Publication Source

The Annals of Probability

Volume

6

Issue

1

Start Page

118

Last Page

127

DOI

10.1214/aop/1176995615

Abstract

If Xi,i=1,2,⋯Xi,i=1,2,⋯ are independent and identically distributed vector valued random variables with distribution F, and S is a class of subsets of Rd, then necessary and sufficient conditions are given for the almost sure convergence of (1/n)Dsn = supAS |(1/n) ∑ 1A(Xi)−F(A)| to zero. The criteria are defined by combinatorial entropies which are given as the time constants of certain subadditive processes. These time constants are estimated, and convergence results for (1/n)DnS obtained, for the classes of algebraic regions, convex sets, and lower layers. These results include the solution to a problem posed by W. Stute.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aop/1176995615#abstract

Comments

At the time of publication, author J. Michael Steele was affiliated with University of British Columbia. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.

Keywords

empirical distribution, subadditive processes, entropy, convex sets, lower layers, algebraic regions

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Date Posted: 27 November 2017

This document has been peer reviewed.