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Limit Theorems for General Empirical Processes

This master thesis is about uniform limit theorems and its main goal is to present a uniform strong law and a uniform weak convergence result for the empirical process indexed by general classes of functions. The topic of uniform versions of the classical limit theorems in probability started in the 1930’s, when Glivenko and then Cantelli proved that the empirical distribution function converges uniformly with probability one to the unknown distribution function of the underlying random variables. This can be considered as a uniform version of the strong law of large numbers. First versions of the central limit theorem for the empirical process were obtained by Doob, Donsker, Kolmogorov and Skorokhod around 1950. In the 1970’s a general theory was started by Vapnik and Cervonenkis, who introduced a new method suitable to treat empirical processes indexed by general function classes. Dudley also made major contributions to this theory. Later Gine and Zinn obtained further refinements. More recently, a weak convergence theory for nonmeasurable mappings has been developed, from which one can infer the limit results for empirical processes. The purpose of this thesis is to given an account of this approach based on the books by Dudley and van der Vaart and Wellner. This version is not the definitive one, but a slightly adapted version, in which the main typos of the first version are corrected and some shortcuts of proofs are implemented, due to the comments of the members of the jury during the presentation.