Central Limit Theorems For Local Emprical Processes Near Boundaries Of Sets
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Central Limit Theorems for Local Empirical Processes Near Boundaries of Sets
We define the local empirical process, based on n i.i.d. random vectors in dimension d, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that for these local empirical processes, indexed by classes of sets that vary with n and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. A continuous mapping theorem for our situation is also derived and some examples are presented.
Weak Convergence and Empirical Processes
Author: Aad van der vaart
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-03-09
This book tries to do three things. The first goal is to give an exposition of certain modes of stochastic convergence, in particular convergence in distribution. The classical theory of this subject was developed mostly in the 1950s and is well summarized in Billingsley (1968). During the last 15 years, the need for a more general theory allowing random elements that are not Borel measurable has become well established, particularly in developing the theory of empirical processes. Part 1 of the book, Stochastic Convergence, gives an exposition of such a theory following the ideas of J. Hoffmann-J!1Jrgensen and R. M. Dudley. A second goal is to use the weak convergence theory background devel oped in Part 1 to present an account of major components of the modern theory of empirical processes indexed by classes of sets and functions. The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, arenaturally viewed as taking values in nonseparable Banach spaces, even in the most elementary cases, and are typically not Borel measurable. Much of the theory presented in Part 2 has previously been scattered in the journal literature and has, as a result, been accessible only to a relatively small number of specialists. In view of the importance of this theory for statis tics, we hope that the presentation given here will make this theory more accessible to statisticians as well as to probabilists interested in statistical applications.
A History of the Central Limit Theorem
Author: Hans Fischer
language: en
Publisher: Springer Science & Business Media
Release Date: 2010-10-08
This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.