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Название: Uniform Central Limit Theorems (Cambridge Studies in Advanced Mathematics)
Автор: Dudley R.
Richard Dudley is a probabilistic and Professor of Mathematics at M.I.T. He is a former editor of the Annals of Probability. This is an advanced probability text. It developed out of courses he gave at M.I.T. and a summer course at St.-Flour in 1982.
Suppose a probability distribution P is defined on the plane. For any half-plane H, defined by a line that splits the plane, the number of points k out of a sample of n falling in the half plane H has a binomial distribution. Normalizing k by subtracting nP(H) (where P(H) is the probability that a randomly selected point falls in H) and dividing by the square root of n leads to a random variable with an asymptotically normal distribution. This is the famous De Moivre - Laplace central limit theorem. This central limit theorem holds simultaneously and uniformly over all half-planes. The uniformity of this result was first proven by M. Donsker. Dudley proves this result in greater generality. Such results are called uniform central limit theorems. There is a general class of sets or functions in more general spaces for which such theorems hold. These sets or functions have been named Donsker classes. Dudley develops the theory in the first 9 chapters. This leads up to the general result for universal Donsker classes in Chapter 10. The two sample case and its application to bootstrapping is given in Chapter 11. Several interesting mathematical results are deferred to the appendices A-I.
This book will be of interests to probabilists, mathematical statisticians and computer scientists working in machine learning theory because it covers the Gine-Zinn bootstrap central limit theorem and provides an extended treatment of Vapnik-Chervonenkis combinatorics among other topics.
Dudley is one of the leading experts on this topic having published numerous articles on it.