## Combinatorial Entropy and Uniform Limit Laws |

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### Contents

THE GENERAL CONVERGENCE THEOREMS | 4 |

SECOND ORDER LIMIT THEOREMS | 15 |

COMBINATORIAL RESULTS AND APPLICATIONS | 31 |

2 other sections not shown

### Common terms and phrases

absolutely continuous distribution AeS i=l AeS(i aperiodic application AS(x AS(XX Borel subsets bounded density chapter choose class of convex class of lower classical column vectors combinatorial constant construct convergence convex Borel convex sets countable set cube define denote distinct columns Doctor of Philosophy E(l l(pJ element empirical distribution function ergodic theory estimate fact finite expectation following property Further we note Glivenko-Cantelli theorem half spaces half-spaces Hence hyperplanes i.i.d. random variables iim i"1 inequality integer interval of F irrational rotation iterated logarithm Kolmogorov entropy L=l i=L large numbers law of large log AS(X1 log2 loglog lower layers Math matrix measurable set measure preserving transformation na)mod observation obtain P(pi P(Xn partition permutation probability space proof of Theorem r)mod Ranga Rao second proof set F Stat stationary ergodic process subadditive submartingale Theorem 2.1 uniformity class Vapnik and Chervonenkis Vapnik-Chervonenkis lemma zero