Empirical Discrepancies and Subadditive Processes

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Statistics Papers
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empirical distribution
subadditive processes
entropy
convex sets
lower layers
algebraic regions
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Steele, J Michael
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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 = supA∈S |(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.

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1978
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The Annals of Probability
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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.
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