Random processes via the combinatorial dimension: introductory notes
نویسندگان
چکیده
This is an informal discussion on one of the basic problems in the theory of empirical processes, addressed in our preprint ”Combinatorics of random processes and sections of convex bodies”, which is available at ArXiV and from our web pages. Arguably, the central problem of the theory of empirical processes is the following: Describe the classes of functions on which the classical limit theorems of probability hold uniformly. Precisely, for a given probability space (Ω, μ), one looks at the sequence of independent samples (Xi) with values in Ω and distributed according to the law μ. Then the problem is to describe the classes F of functions f : Ω → R for which the sequence of real valued random variables f(Xi) satisfies the classical limit theorems of probability uniformly over f ∈ F . The two classical limit theorems we have in mind are the law of large numbers, for which such classes are called Glivenko-Cantelli, and the central limit theorem, which gives rise to Donsker classes. In practice one does not know the law μ according to which the samples Xi are drawn. Thus a particularly vital question is to know what classes are GlivenkoCantelli or Donsker for every μ. Such classes are called universal or even uniform if the convergence in the corresponding limit theorems is uniform over all μ. We refer the reader to Chapter 14.3 of the book of Ledoux and Talagrand [LT] for a brief introduction to the theory of empirical processes and to the book of Dudley [Du 99] for a comprehensive account. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA; e-mail: [email protected] Deptartment of Mathematics, University of California, Davis, CA 95616, USA; e-mail: [email protected]
منابع مشابه
CLUSTER ALGEBRAS AND CLUSTER CATEGORIES
These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...
متن کاملProbability on Trees: An Introductory Climb
2 1 Preface These notes are based on Lectures delivered at the Saint Flour Summer School in July 1997. Accurate notes were taken by Dimitris Gatzouras, who also edited them. Revisions by David Levin and myself led to the current form of the notes. I hope that they are useful to probabilists and graduate students as an introduction to the subject; a more complete account is in the forthcoming bo...
متن کاملA Few Notes on Statistical Learning Theory
2 Glivenko-Cantelli Classes 5 2.1 The classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 The symmetrization procedure . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Covering numbers and complexity estimates . . . . . . . . . . . . . . 9 2.2 Combinatorial parameters and covering numbers . . . . . . . . . . . . . . . 12 2.2.1 Uniform entropy and the VC dimen...
متن کاملA Deterministic Multiple Key Space Scheme for Wireless Sensor Networks via Combinatorial Designs
The establishing of a pairwise key between two nodes for encryption in a wireless sensor network is a challenging issue. To do this, we propose a new deterministic key pre-distribution scheme which has modified the multiple key space scheme (MKSS). In the MKSS, the authors define two random parameters to make better resilience than existing schemes. Instead of a random selection of these parame...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004