(Ebook PDF) Gaussian Measures in Hilbert Space Construction and Properties 1st edition by Alexander Kukush 1119686725 9781119686729 full chapters

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ISBN-10 :  1119686725 

ISBN-13 :  9781119686729

Author : Alexander Kukush 

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 Fernique’s 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.

Gaussian Measures in Hilbert Space: Construction and Properties 1st Table of contents:

1 Gaussian Measures in Euclidean Space
1.1. The change of variables formula
1.2. Invariance of Lebesgue measure
1.3. Absence of invariant measure in infinite-dimensional Hilbert space
1.4. Random vectors and their distributions
1.5. Gaussian vectors and Gaussian measures
2 Gaussian Measure in l2 as a Product Measure
2.1. Space ℝ∞
2.2. Product measure in ℝ∞
2.3. Standard Gaussian measure in ℝ∞
2.4. Construction of Gaussian measure in l2
3 Borel Measures in Hilbert Space
3.1. Classes of operators in H
3.2. Pettis and Bochner integrals
3.3. Borel measures in Hilbert space
4 Construction of Measure by its Characteristic Functional
4.1. Cylindrical sigma-algebra in normed space
4.2. Convolution of measures
4.3. Properties of characteristic functionals in H
4.4. S-topology in H
4.5. Minlos–Sazonov theorem
5 Gaussian Measure of General Form
5.1. Characteristic functional of Gaussian measure
5.2. Decomposition of Gaussian measure and Gaussian random element
5.3. Support of Gaussian measure and its invariance
5.4. Weak convergence of Gaussian measures
5.5. Exponential moments of Gaussian measure in normed space
6 Equivalence and Singularity of Gaussian Measures
6.1. Uniformly integrable sequences
6.2. Kakutani’s dichotomy for product measures on ℝ∞
6.3. Feldman–Hájek dichotomy for Gaussian measures on H
6.4. Applications in statistics
7 Solutions
7.1. Solutions for Chapter 1
7.2. Solutions for Chapter 2
7.3. Solutions for Chapter 3
7.4. Solutions for Chapter 4
7.5. Solutions for Chapter 5
7.6. Solutions for Chapter 6

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Gaussian Measures,Hilbert Space,Construction,Properties,Alexander Kukush