Gaussian Measures In Hilbert Space

Author : Alexander Kukush
Publisher : John Wiley & Sons
Page : 272 pages
File Size : 50,8 Mb
Release : 2020-02-26
Category : Mathematics
ISBN : 9781786302670

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Gaussian Measures in Hilbert Space by Alexander Kukush Pdf

At the nexus of probability theory, geometry and statistics, a Gaussian measure is constructed on a Hilbert space in two ways: as a product measure and via a characteristic functional based on Minlos-Sazonov theorem. As such, it can be utilized for obtaining results for topological vector spaces. Gaussian Measures contains the proof for Ferniques theorem and its relation to exponential moments in Banach space. Furthermore, the fundamental Feldman-Hájek dichotomy for Gaussian measures in Hilbert space is investigated. Applications in statistics are also outlined. In addition to chapters devoted to measure theory, this book highlights problems related to Gaussian measures in Hilbert and Banach spaces. Borel probability measures are also addressed, with properties of characteristic functionals examined and a proof given based on the classical Banach–Steinhaus theorem. Gaussian Measures is suitable for graduate students, plus advanced undergraduate students in mathematics and statistics. It is also of interest to students in related fields from other disciplines. Results are presented as lemmas, theorems and corollaries, while all statements are proven. Each subsection ends with teaching problems, and a separate chapter contains detailed solutions to all the problems. With its student-tested approach, this book is a superb introduction to the theory of Gaussian measures on infinite-dimensional spaces.

Author : H.-H. Kuo
Publisher : Springer
Page : 230 pages
File Size : 48,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540375081

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Gaussian Measures in Banach Spaces by H.-H. Kuo Pdf

Author : Vladimir I. Bogachev
Publisher : American Mathematical Soc.
Page : 433 pages
File Size : 45,9 Mb
Release : 2015-01-26
Category : Electronic
ISBN : 9781470418694

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Gaussian Measures by Vladimir I. Bogachev Pdf

This book gives a systematic exposition of the modern theory of Gaussian measures. It presents with complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Suitable for use as a graduate text and/or a reference work, this volume contains many examples, exercises, and an extensive bibliography. It brings together many results that have not appeared previously in book form.

Author : Yaozhong Hu
Publisher : World Scientific
Page : 484 pages
File Size : 52,7 Mb
Release : 2016-08-30
Category : Mathematics
ISBN : 9789813142190

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Analysis on Gaussian Spaces by Yaozhong Hu Pdf

Analysis of functions on the finite dimensional Euclidean space with respect to the Lebesgue measure is fundamental in mathematics. The extension to infinite dimension is a great challenge due to the lack of Lebesgue measure on infinite dimensional space. Instead the most popular measure used in infinite dimensional space is the Gaussian measure, which has been unified under the terminology of "abstract Wiener space". Out of the large amount of work on this topic, this book presents some fundamental results plus recent progress. We shall present some results on the Gaussian space itself such as the Brunn–Minkowski inequality, Small ball estimates, large tail estimates. The majority part of this book is devoted to the analysis of nonlinear functions on the Gaussian space. Derivative, Sobolev spaces are introduced, while the famous Poincaré inequality, logarithmic inequality, hypercontractive inequality, Meyer's inequality, Littlewood–Paley–Stein–Meyer theory are given in details. This book includes some basic material that cannot be found elsewhere that the author believes should be an integral part of the subject. For example, the book includes some interesting and important inequalities, the Littlewood–Paley–Stein–Meyer theory, and the Hörmander theorem. The book also includes some recent progress achieved by the author and collaborators on density convergence, numerical solutions, local times.

Author : Daniel W. Stroock
Publisher : Springer Nature
Page : 152 pages
File Size : 52,7 Mb
Release : 2023-02-15
Category : Mathematics
ISBN : 9783031231223

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Gaussian Measures in Finite and Infinite Dimensions by Daniel W. Stroock Pdf

This text provides a concise introduction, suitable for a one-semester special topicscourse, to the remarkable properties of Gaussian measures on both finite and infinitedimensional spaces. It begins with a brief resumé of probabilistic results in which Fourieranalysis plays an essential role, and those results are then applied to derive a few basicfacts about Gaussian measures on finite dimensional spaces. In anticipation of the analysisof Gaussian measures on infinite dimensional spaces, particular attention is given to those/divproperties of Gaussian measures that are dimension independent, and Gaussian processesare constructed. The rest of the book is devoted to the study of Gaussian measures onBanach spaces. The perspective adopted is the one introduced by I. Segal and developedby L. Gross in which the Hilbert structure underlying the measure is emphasized.The contents of this book should be accessible to either undergraduate or graduate/divstudents who are interested in probability theory and have a solid background in Lebesgueintegration theory and a familiarity with basic functional analysis. Although the focus ison Gaussian measures, the book introduces its readers to techniques and ideas that haveapplications in other contexts.

Author : Svante Janson
Publisher : Cambridge University Press
Page : 358 pages
File Size : 43,5 Mb
Release : 1997-06-12
Category : Mathematics
ISBN : 9780521561280

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Gaussian Hilbert Spaces by Svante Janson Pdf

This book treats the very special and fundamental mathematical properties that hold for a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, other parts of mathematics, statistics and theoretical physics. The emphasis throughout this book is on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.

Author : Alain Guichardet
Publisher : Springer
Page : 203 pages
File Size : 48,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540374558

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Symmetric Hilbert Spaces and Related Topics by Alain Guichardet Pdf

Author : Timo Koski
Publisher : Unknown
Page : 128 pages
File Size : 40,7 Mb
Release : 1984
Category : Electronic
ISBN : 9516490107

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Statistical Spaces of Gaussian Measures on a Hilbert Space and Their Ellipsoids of Variance by Timo Koski Pdf

Author : A. A. Dorogovtsev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 116 pages
File Size : 43,5 Mb
Release : 2019-01-14
Category : Mathematics
ISBN : 9783110618143

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Stochastic Analysis and Random Maps in Hilbert Space by A. A. Dorogovtsev Pdf

This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. In this book, objects from the Gaussian case are considered. Therefore, it is useful to consider all random variables and elements as functionals from the Wiener process or its formal derivative, i.e. white noise. The book consists of five chapters. The first chapter is devoted to stochastic calculus and its main goal is to prepare the tools for solving stochastic equations. In the second chapter the structure of stochastic equations, mainly the structure of Gaussian strong linear operators, is studied. In chapter 3 the definition of the action of the stochastic operator on random elements in considered. Chapter 4 deals with the mathematical models in which the notions of stochastic calculus arise and in the final chapter the equation with random operators is considered.

Author : I͡Uriĭ Anatolʹevich Rozanov
Publisher : American Mathematical Soc.
Page : 172 pages
File Size : 52,8 Mb
Release : 1971
Category : Distribution (Probability theory)
ISBN : 0821830082

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Infinite-Dimensional Gaussian Distributions by I͡Uriĭ Anatolʹevich Rozanov Pdf

Author : Giuseppe Da Prato
Publisher : Springer Science & Business Media
Page : 217 pages
File Size : 40,5 Mb
Release : 2006-08-25
Category : Mathematics
ISBN : 9783540290216

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An Introduction to Infinite-Dimensional Analysis by Giuseppe Da Prato Pdf

Based on well-known lectures given at Scuola Normale Superiore in Pisa, this book introduces analysis in a separable Hilbert space of infinite dimension. It starts from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate basic stochastic dynamical systems and Markov semi-groups, paying attention to their long-time behavior.

Author : Leonard Gross
Publisher : American Mathematical Soc.
Page : 62 pages
File Size : 40,7 Mb
Release : 1963
Category : Harmonic analysis
ISBN : 9780821812464

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Harmonic Analysis on Hilbert Space by Leonard Gross Pdf

Author : Mikhail Lifshits
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 52,8 Mb
Release : 2012-01-13
Category : Mathematics
ISBN : 9783642249389

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Lectures on Gaussian Processes by Mikhail Lifshits Pdf

Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.​

Author : Howard L. Weinert
Publisher : Unknown
Page : 680 pages
File Size : 41,9 Mb
Release : 1982
Category : Mathematics
ISBN : STANFORD:36105031984888

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Reproducing Kernel Hilbert Spaces by Howard L. Weinert Pdf

Author : Giuseppe Da Prato,Jerzy Zabczyk
Publisher : Cambridge University Press
Page : 397 pages
File Size : 52,7 Mb
Release : 2002-07-25
Category : Mathematics
ISBN : 9781139433433

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Second Order Partial Differential Equations in Hilbert Spaces by Giuseppe Da Prato,Jerzy Zabczyk Pdf

State of the art treatment of a subject which has applications in mathematical physics, biology and finance. Includes discussion of applications to control theory. There are numerous notes and references that point to further reading. Coverage of some essential background material helps to make the book self contained.