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February, 1978 Empirical Discrepancies and Subadditive Processes
J. Michael Steele
Ann. Probab. 6(1): 118-127 (February, 1978). DOI: 10.1214/aop/1176995615


If $X_i, i = 1,2,\cdots$ are independent and identically distributed vector valued random variables with distribution $F$, and $S$ is a class of subsets of $R^d$, then necessary and sufficient conditions are given for the almost sure convergence of $(1/n)D_n^s = \sup_{A\in S} |(1/n) \sum 1_A(X_i) - F(A)|$ 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 $(1/n)D_n^S$ 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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J. Michael Steele. "Empirical Discrepancies and Subadditive Processes." Ann. Probab. 6 (1) 118 - 127, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0374.60037
MathSciNet: MR464379
Digital Object Identifier: 10.1214/aop/1176995615

Primary: 60F15
Secondary: 60G99

Keywords: algebraic regions , convex sets , Empirical distribution , Entropy , lower layers , Subadditive processes

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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