Empirical Discrepancies and Subadditive Processes

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.