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Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012.
The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.
Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012.
The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.
First book devoted to the topic of super concentration, chaos and multiple valleys
Presents a wide array of examples on the subject
Integrates new concepts and gives a systematic account of the history and development of the features of random objects
Includes supplementary material: [...]
Preface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand's method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.
| Erscheinungsjahr: | 2016 |
|---|---|
| Fachbereich: | Wahrscheinlichkeitstheorie |
| Genre: | Mathematik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Taschenbuch |
| Seiten: | 168 |
| Reihe: | Springer Monographs in Mathematics |
| Inhalt: |
ix
156 S. |
| ISBN-13: | 9783319352282 |
| ISBN-10: | 3319352288 |
| Sprache: | Englisch |
| Ausstattung / Beilage: | Paperback |
| Einband: | Kartoniert / Broschiert |
| Autor: | Chatterjee, Sourav |
| Auflage: | Softcover reprint of the original 1st ed. 2014 |
| Hersteller: |
Springer International Publishing
Springer International Publishing AG Springer Monographs in Mathematics |
| Maße: | 235 x 155 x 10 mm |
| Von/Mit: | Sourav Chatterjee |
| Erscheinungsdatum: | 23.08.2016 |
| Gewicht: | 0,265 kg |
First book devoted to the topic of super concentration, chaos and multiple valleys
Presents a wide array of examples on the subject
Integrates new concepts and gives a systematic account of the history and development of the features of random objects
Includes supplementary material: [...]
Preface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand's method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.
| Erscheinungsjahr: | 2016 |
|---|---|
| Fachbereich: | Wahrscheinlichkeitstheorie |
| Genre: | Mathematik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Taschenbuch |
| Seiten: | 168 |
| Reihe: | Springer Monographs in Mathematics |
| Inhalt: |
ix
156 S. |
| ISBN-13: | 9783319352282 |
| ISBN-10: | 3319352288 |
| Sprache: | Englisch |
| Ausstattung / Beilage: | Paperback |
| Einband: | Kartoniert / Broschiert |
| Autor: | Chatterjee, Sourav |
| Auflage: | Softcover reprint of the original 1st ed. 2014 |
| Hersteller: |
Springer International Publishing
Springer International Publishing AG Springer Monographs in Mathematics |
| Maße: | 235 x 155 x 10 mm |
| Von/Mit: | Sourav Chatterjee |
| Erscheinungsdatum: | 23.08.2016 |
| Gewicht: | 0,265 kg |
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