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# Empirical Discrepancies and Subadditive Processes

J. Michael Steele
The Annals of Probability
Vol. 6, No. 1 (Feb., 1978), pp. 118-127
Stable URL: http://www.jstor.org/stable/2242865
Page Count: 10
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## Abstract

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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