Stable domains of attraction for empirical processes on Vapnik-Červonenkis classes of functions

Abstract

The main goal of this dissertation is to extend Alexander's (1987) central limit theorem for empirical processes on Vapnik-Cervonenkis classes of functions to the case with non-Gaussian stable Radon limit. The dissertation is divided into four chapters. The first one presents a review on the theory of empirical processes; it also contains the definitions and previous results needed in the following chapters. In the second chapter, we study the entropy of Vapnik-Cervonenkis classes of functions and we prove an exponential inequality similar to one in Alexander (1987). The main results are established in Chapter three. In the first section we give a different proof of Alexander's central limit theorem for empirical processes on Vapnik-Cervonenkis classes of functions. The second section contains sufficient conditions for the process {f(X): f [epsilon] F} to be in the normal domain of attraction of a p-stable Radon measure in ℓ[superscript infinity](F),1 [less than or equal to] p < 2. We also give necessary and sufficient conditions in the case of Vapnik-Cervonenkis classes of functions. The corresponding weak laws of large numbers and results about stochastic boundedness are also considered. The last chapter presents several applications of the previous results. Theorem 4.2 gives a rate of convergence for a result of Pollard (1981b) in clustering analysis. We also show how Theorem 3.3 implies a central limit theorem in C(K) (Araujo and Gine (1979), improved by Marcus and Pisier (1984)) and we apply the weak law of large numbers to weighted empirical processes.