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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.
Contents
Foreword ix
Preface xiii
Introduction xv
Abbreviations and Notation xix
Chapter 1. Gaussian Measures in Euclidean Space 1
1.1. The change of variables formula 1
1.2. Invariance of Lebesgue measure 4
1.3. Absence of invariant measure in infinite-dimensional Hilbert space 9
1.4. Random vectors and their distributions 10
1.4.1. Random variables 11
1.4.2. Random vectors 12
1.4.3. Distributions of random vectors 14
1.5. Gaussian vectors and Gaussian measures 17
1.5.1. Characteristic functions of Gaussian vectors 17
1.5.2. Expansion of Gaussian vector 20
1.5.3. Support of Gaussian vector 22
1.5.4. Gaussian measures in Euclidean space 23
Chapter 2. Gaussian Measure in l2 as a Product Measure 27
2.1. Space R∞ 27
2.1.1. Metric on R∞ 27
2.1.2. Borel and cylindrical sigma-algebras coincide 30
2.1.3. Weighted l2 space 31
2.2. Product measure in R∞ 34
2.2.1. Kolmogorov extension theorem 34
2.2.2. Construction of product measure on B(R∞) 36
2.2.3. Properties of product measure 38
2.3. Standard Gaussian measure in R∞ 42
2.3.1. Alternative proof of the second part of theorem 2.4 45
2.4. Construction of Gaussian measure in l2 46
Chapter 3. Borel Measures in Hilbert Space 51
3.1. Classes of operators in H 51
3.1.1. Hilbert-Schmidt operators 52
3.1.2. Polar decomposition 55
3.1.3. Nuclear operators 57
3.1.4. S-operators 62
3.2. Pettis and Bochner integrals 68
3.2.1. Weak integral 68
3.2.2. Strong integral 69
3.3. Borel measures in Hilbert space 75
3.3.1. Weak and strong moments 75
3.3.2. Examples of Borel measures 78
3.3.3. Boundedness of moment form 83
Chapter 4. Construction of Measure by its Characteristic Functional 89
4.1. Cylindrical sigma-algebra in normed space 89
4.2. Convolution of measures 93
4.3. Properties of characteristic functionals in H 96
4.4. S-topology in H 99
4.5. Minlos-Sazonov theorem 102
Chapter 5. Gaussian Measure of General Form 111
5.1. Characteristic functional of Gaussian measure 111
5.2. Decomposition of Gaussian measure and Gaussian random element 114
5.3. Support of Gaussian measure and its invariance 117
5.4. Weak convergence of Gaussian measures 125
5.5. Exponential moments of Gaussian measure in normed space 129
5.5.1. Gaussian measures in normed space 129
5.5.2. Fernique's theorem 133
Chapter 6. Equivalence and Singularity of Gaussian Measures 143
6.1. Uniformly integrable sequences 143
6.2. Kakutani's dichotomy for product measures on R∞ 145
6.2.1. General properties of absolutely continuous measures 145
6.2.2. Kakutani's theorem for product measures 148
6.2.3. Dichotomy for Gaussian product measures 152
6.3. Feldman-Hájek dichotomy for Gaussian measures on H 155
6.3.1. The case where Gaussian measures have equal correlation operators 155
6.3.2. Necessary conditions for equivalence of Gaussian measures 158
6.3.3. Criterion for equivalence of Gaussian measures 165
6.4. Applications in statistics 169
6.4.1. Estimation and hypothesis testing for mean of Gaussian random element 169
6.4.2. Estimation and hypothesis testing for correlation operator of centered Gaussian random element 173
Chapter 7. Solutions 179
7.1. Solutions for Chapter 1 179
7.2. Solutions for Chapter 2 193
7.2.1. Generalized Kolmogorov extension theorem 196
7.3. Solutions for Chapter 3 202
7.4. Solutions for Chapter 4 211
7.5. Solutions for Chapter 5 217
7.6. Solutions for Chapter 6 227
Summarizing Remarks 235
References 239
Index 241
