Gaussian Measures in Hilbert Space Construction and Properties

Foreword ix

Preface xiii

Introduction xv

Abbreviations and Notation xix

Chapter 1. Gaussian Measures in Euclidean Space1

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 l₂as a Product Measure27

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 Space51

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 Functional89

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 Form111

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 Measures143

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. Solutions179

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