New isoperimetric inequalities and random process techniques have recently appeared at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (boundedness and continuity of random processes, integrability and limit theorems for vector valued random variables,...) and of some of their links to Geometry of Banach spaces. Its purpose is to present some of the main aspects of this theory, from the foundations to the latest develop- ments, treated with the most recent and updated tools. In particular, the most important features are the sys- tematic use of isoperimetry and related concentration of measure phenomena (to study integrability and limit theorems for vector valued random variables), and recent abstract random process techniques (entropy and majorizing measures). Some examples of these probabilistic ideas to classical Banach space theory complete this exposition.
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This book gives an excellent, almost complete account of the whole subject of probability in Banach spaces, a branch of probability theory that has undergone vigorous development... There is no doubt in the reviewer's mind that this book [has] become a classic. MathSciNet As the authors state, "this book tries to present some of the main aspects of the theory of probability in Banach spaces, from the foundation of the topic to the latest developments and current research questions''. The authors have succeeded admirably… This very comprehensive book develops a wide variety of the methods existing … in probability in Banach spaces. … It [has] become an event for mathematicians… Zentralblatt MATH
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Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.
--This text refers to an alternate
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5.0 out of 5 stars
A must in modern Probability Theory,
This review is from: Probability in Banach Spaces (Hardcover)
This book is the standard for all who seek detailed information about stochastics in Banach spaces. Written from two of the top researchers in this area, it provides a broad area of topics, a concise treatment of deep results and exhaustive references. Sadly enough, it is out of print now, but I hope for a 2nd edition.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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orthogaussian sequence, majorizing measure conditions, isoperimetric approach, strong second moment, cotype properties, gixi converges, zero real valued random variables, sure boundedness, empirical process methods, subgaussian inequality, closed graph argument, majorizing measures, subgaussian processes, metric entropy conditions, isoperimetric methods, ball criterion, space valued random variables, independent mean zero random variables, finite cotype, strong limit theorems, stable random vectors, symmetric random variables, central limit property, vector valued coefficients, stable random variables Key Phrases - Capitalized Phrases (CAPs): (learn more)
Gaussian Radon, Gaussian Random Variables, The Law of the Iterated Logarithm, Regularity of Gaussian Processes, Some Probabilistic Statements, Banach Space Theory, Regularity of Random Processes, Some Central Limit Theorems, Examples of Laws of Large Numbers, Some Isoperimetric Inequalities, Examples of Applications, Probability Theory, Integrability of Gaussian Chaos, Using Lemma, Kolmogorov's Law of the Iterated Logarithm, Integrability of Rademacher Chaos, Harmonic Analysis, Vapnik-Chervonenkis Classes of Sets, Geometry of Banach, Gaussian Fourier, Comparison Theorems Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
orthogaussian sequence, majorizing measure conditions, isoperimetric approach, strong second moment, cotype properties, gixi converges, zero real valued random variables, sure boundedness, empirical process methods, subgaussian inequality, closed graph argument, majorizing measures, subgaussian processes, metric entropy conditions, isoperimetric methods, ball criterion, space valued random variables, independent mean zero random variables, finite cotype, strong limit theorems, stable random vectors, symmetric random variables, central limit property, vector valued coefficients, stable random variables Key Phrases - Capitalized Phrases (CAPs): (learn more)
Gaussian Radon, Gaussian Random Variables, The Law of the Iterated Logarithm, Regularity of Gaussian Processes, Some Probabilistic Statements, Banach Space Theory, Regularity of Random Processes, Some Central Limit Theorems, Examples of Laws of Large Numbers, Some Isoperimetric Inequalities, Examples of Applications, Probability Theory, Integrability of Gaussian Chaos, Using Lemma, Kolmogorov's Law of the Iterated Logarithm, Integrability of Rademacher Chaos, Harmonic Analysis, Vapnik-Chervonenkis Classes of Sets, Geometry of Banach, Gaussian Fourier, Comparison Theorems Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
55
books
cite this book:
See all 55 books citing this book
- High Dimensional Probability II (Progress in Probability) by Evarist Giné on 4 pages
- Probability in Banach Spaces, 9 (Progress in Probability) by Jorgen Hoffmann-Jorgensen in Front Matter, page 206, and page 323
- Stochastic Inequalities and Applications (Progress in Probability) by Evariste Giné on page 75, page 211, and page 213
- Uniform Central Limit Theorems (Cambridge Studies in Advanced Mathematics) by R. M. Dudley on page 232, and page 313
- Mathematical Aspects of Spin Glasses and Neural Networks (Progress in Probability) by Anton Bovier on page 87, and page 149
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