The concentration of measure phenomenon
Author(s)
Bibliographic Information
The concentration of measure phenomenon
（Mathematical surveys and monographs, v. 89）
American Mathematical Society, c2001
- : pbk
Available at / 50 libraries
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Hokkaido University, Library, Graduate School of Science, Faculty of Science and School of Science図書
DC21:519.2/L4992070548207
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LCCN:2001041310
Includes bibliographical references (p. 171-179) and index
Description and Table of Contents
Description
The observation of the concentration of measure phenomenon is inspired by isoperimetric inequalities. A familiar example is the way the uniform measure on the standard sphere $Sn$ becomes concentrated around the equator as the dimension gets large. This property may be interpreted in terms of functions on the sphere with small oscillations, an idea going back to Levy. The phenomenon also occurs in probability, as a version of the law of large numbers, due to Emil Borel. This book offers the basic techniques and examples of the concentration of measure phenomenon. The concentration of measure phenomenon was put forward in the early seventies by V. Milman in the asymptotic geometry of Banach spaces. It is of powerful interest in applications in various areas, such as geometry, functional analysis and infinite-dimensional integration, discrete mathematics and complexity theory, and probability theory.
Particular emphasis is on geometric, functional, and probabilistic tools to reach and describe measur The book presents concentration functions and inequalities, isoperimetric and functional examples, spectrum and topological applications, product measures, entropic and transportation methods, as well as aspects of M. Talagrand's deep investigation of concentration in product spaces and its application in discrete mathematics and probability theory, supremum of Gaussian and empirical processes, spin glass, random matrices, etc. Prerequisites are a basic background in measure theory, functional analysis, and probability theory.
Table of Contents
- Concentration functions and inequalities
- Isoperimetric and functional examples
- Concentration and geometry
- Concentration in product spaces
- Entropy and concentration
- Transportation cost inequalities
- Sharp bounds of Gaussian and empirical processes
- Selected applications
- References
- Index
by "Nielsen BookData"
